Coordinate Geometry — Extra Practice

Competitive Mathematics · Geometry

6 Part10 worked examples328 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

A place on a map is given by a row and a column. A point on a plane is located the same way by two numbers $(x, y)$ — the first says how far left or right of the origin, the second how far up or down. Once points are numbers, the geometry of distance, area and lines becomes algebra and can be solved even without drawing.

∑ Formula

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}; मध्य - बिंदु = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}); m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

The three base formulas

Points, distance and division

The arithmetic of points on a plane

Quadrants, reflection, distance, equidistant points, mid-point, section formula and centroid.

Axes, quadrants and reflection

The signs tell you where the point is

The coordinate axes, the four quadrants and reflection in the axes.

◭ Diagram

The four quadrants: I

(+, +)

, II

(-, +)

, III

(-, -)

, IV

(+, -)

Abscissa ( $x$ ) = distance from the Y-axis (right $+$ , left $-$ ); ordinate ( $y$ ) = distance from the X-axis (up $+$ , down $-$ ).
A point on the X-axis is $(x, 0)$ ; on the Y-axis $(0, y)$ ; the origin $(0, 0)$ . Points on the axes belong to no quadrant.
Reflection: in the X-axis $(x, - y)$ ; in the Y-axis $(- x, y)$ ; in the origin $(- x, - y)$ ; in the line $y = x$ $(y, x)$ .

Example 1 · Basic

In which quadrants do the points

(- 6, - 2)

and

(3, - 7)

lie?

Solution, step by step

1 $(- 6, - 2)$ : both negative ⟹ 3rd quadrant.
2 $(3, - 7)$ : $x$ positive, $y$ negative ⟹ 4th quadrant

Answer 3rd and 4th

BasicTry it yourself first

Where does the point (-3, -8) lie?

BasicTry it yourself first

Where does the point (6, -2) lie?

BasicTry it yourself first

Where does the point (-4, 9) lie?

StandardTry it yourself first

Where does the point (0, -5) lie?

StandardTry it yourself first

Where does the point (7, 0) lie?

StandardTry it yourself first

Where does the point (0, 0) lie?

StandardTry it yourself first

Where does the point (2, 11) lie?

AdvancedTry it yourself first

Where does the point (-12, 5) lie?

AdvancedTry it yourself first

Where does the point (0, 8) lie?

BasicTry it yourself first

Find the reflection of the point (3, -5) in the X-axis.

BasicTry it yourself first

Find the reflection of the point (-4, 6) in the Y-axis.

StandardTry it yourself first

Find the reflection of the point (2, 7) in the origin.

StandardTry it yourself first

Find the reflection of the point (3, 8) in the line

y = x

StandardTry it yourself first

Find the reflection of the point (-5, -9) in the X-axis.

AdvancedTry it yourself first

Find the reflection of the point (-6, 11) in the origin.

AdvancedTry it yourself first

Find the reflection of the point (-7, 4) in the line

y = x

AdvancedTry it yourself first

Find the reflection of the point (8, -13) in the Y-axis.

Distance and equidistant points

Pythagoras in disguise

The distance formula, unknown coordinates and points equally far from two given points.

∑ Formula

A B = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}; OP = x^{2} + y^{2}

Distance formula (distance from the origin is the special case)

The distance formula is just Pythagoras: the difference in $x$ and in $y$ are the two sides of a right triangle.
Unknown coordinate: square the distance to make an equation; $(x - a)^{2} = r^{2}$ gives $x = a \pm r$ — two answers.
The locus of points equally far from two points is the perpendicular bisector of the segment joining them. Writing $P A^{2} = P B^{2}$ , the $x^{2}, y^{2}$ terms cancel, leaving a line.
Equidistant point on an axis: take $(p, 0)$ or $(0, p)$ and solve $P A^{2} = P B^{2}$ .

Example 2 · Basic

Find the distance between

A (3, 4)

and

B (- 2, - 8)

Solution, step by step

1Differences: $Δ x = - 2 - 3 = - 5$ , $Δ y = - 8 - 4 = - 12$ .
2 $A B = 25 + 144 = 169 = 13$ units

Answer 13 units

Example 3 · Standard

Find the point on the Y-axis equidistant from

(5, 2)

and

(- 3, 6)

Solution, step by step

1Let $P (0, q)$ : $25 + (q - 2)^{2} = 9 + (q - 6)^{2}$ .
2 $25 - 4 q + 4 = 9 - 12 q + 36$ ⟹ $8 q = 16$ ⟹ $q = 2$ ; the point is $(0, 2)$

Answer $(0, 2)$

BasicTry it yourself first

Find the distance between the points (1, 2) and (4, 6).

BasicTry it yourself first

Find the distance between the points (-2, 3) and (4, -5).

StandardTry it yourself first

Find the distance between the points (5, 1) and (-7, 6).

StandardTry it yourself first

Find the distance between the points (0, 0) and (3, 7).

StandardTry it yourself first

Find the distance between the points (-1, 2) and (3, 4).

AdvancedTry it yourself first

Find the distance between the points (4, -3) and (-2, 1).

AdvancedTry it yourself first

Find the distance between the points (-6, -2) and (9, 6).

AdvancedTry it yourself first

Find the distance between the points (7, 3) and (-17, -4).

BasicTry it yourself first

Find the distance of the point (6, 8) from the origin.

BasicTry it yourself first

Find the distance of the point (5, -12) from the origin.

StandardTry it yourself first

Find the distance of the point (8, -15) from the origin.

StandardTry it yourself first

Find the distance of the point (2, 6) from the origin.

AdvancedTry it yourself first

Find the distance of the point (-7, 24) from the origin.

AdvancedTry it yourself first

Find the distance of the point (-4, -7) from the origin.

StandardTry it yourself first

The distance between the points

(x, 7)

and (2, 3) is 5 units. Find the possible values of

x

StandardTry it yourself first

The distance between the points

(x, 4)

and (5, -2) is 10 units. Find the possible values of

x

AdvancedTry it yourself first

The distance between the points

(x, - 5)

and (-4, 1) is 10 units. Find the possible values of

x

AdvancedTry it yourself first

The distance between the points

(x, 14)

and (3, 2) is 13 units. Find the possible values of

x

StandardTry it yourself first

Find the equation of the locus of points equidistant from (1, 2) and (5, 6).

StandardTry it yourself first

Find the equation of the locus of points equidistant from (0, 2) and (6, -2).

AdvancedTry it yourself first

Find the equation of the locus of points equidistant from (3, -1) and (-1, 5).

AdvancedTry it yourself first

Find the equation of the locus of points equidistant from (-2, 4) and (4, 0).

StandardTry it yourself first

Find the point on the X-axis equidistant from (1, 3) and (5, 1).

StandardTry it yourself first

Find the point on the X-axis equidistant from (3, 5) and (7, 1).

AdvancedTry it yourself first

Find the point on the X-axis equidistant from (1, 2) and (7, 4).

AdvancedTry it yourself first

Find the point on the X-axis equidistant from (1, 6) and (5, 2).

Mid-point, section and centroid

Averages and weighted averages

The mid-point formula, internal and external division, the ratio in which an axis divides a segment, and the centroid.

Quantity	Formula
Mid-point	$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Internal division $m : n$	$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$
External division $m : n$	$(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n})$
Centroid of a triangle	$(\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})$

The mid-point is the special case $m = n$ . The other end $= 2 \times$ mid-point $-$ first end.
The ratio in which an axis cuts a segment: on the X-axis $y = 0$ , so $m : n = - y_{1} : y_{2}$ . A positive ratio is internal, a negative one external.
The triangle of mid-points and the original have the same centroid; so if mid-points are given, their average is the centroid.
Third vertex: $3 \times$ centroid $-$ the two known vertices. In a parallelogram $D = A + C - B$ (the diagonals share a mid-point).

Example 4 · Basic

Find the point dividing the segment joining

A (- 1, 7)

and

B (4, - 3)

internally in the ratio

2 : 3

Solution, step by step

1 $x = \frac{2 ( 4 ) + 3 ( - 1 )}{5} = \frac{5}{5} = 1$ .
2 $y = \frac{2 ( - 3 ) + 3 ( 7 )}{5} = \frac{15}{5} = 3$ ; the point is $(1, 3)$

Answer $(1, 3)$

Example 5 · Standard

The mid-points of the sides of a triangle are

(4, 1)

(7, 5)

and

(1, 6)

. Find the vertices.

Solution, step by step

1Vertex $=$ sum of two mid-points $-$ the third: $(7 + 1 - 4, 5 + 6 - 1) = (4, 10)$ .
2Likewise $(4 + 1 - 7, 1 + 6 - 5) = (- 2, 2)$ and $(4 + 7 - 1, 1 + 5 - 6) = (10, 0)$

Answer $(4, 10), (- 2, 2), (10, 0)$

BasicTry it yourself first

Find the mid-point of the segment joining (2, 3) and (8, 7).

BasicTry it yourself first

Find the mid-point of the segment joining (-4, 6) and (10, -2).

StandardTry it yourself first

Find the mid-point of the segment joining (5, -3) and (-9, 8).

AdvancedTry it yourself first

Find the mid-point of the segment joining (-11, -6) and (7, 13).

StandardTry it yourself first

The mid-point of segment AB is (3, 5) and A is (-1, 2). Find the coordinates of B.

StandardTry it yourself first

The mid-point of segment AB is (2, -1) and A is (6, 4). Find the coordinates of B.

AdvancedTry it yourself first

The mid-point of segment AB is (-3, 4) and A is (5, -7). Find the coordinates of B.

AdvancedTry it yourself first

The mid-point of segment AB is (0, 6) and A is (-8, 3). Find the coordinates of B.

StandardTry it yourself first

Find the point that divides the segment joining A(1, 2) and B(7, 8) internally in the ratio 2 : 1.

StandardTry it yourself first

Find the point that divides the segment joining A(-2, 1) and B(8, 6) internally in the ratio 3 : 2.

AdvancedTry it yourself first

Find the point that divides the segment joining A(2, -3) and B(7, 12) internally in the ratio 3 : 2.

StandardTry it yourself first

Find the point that divides the segment joining A(2, 1) and B(6, 5) externally in the ratio 3 : 1.

AdvancedTry it yourself first

Find the point that divides the segment joining A(-1, 3) and B(3, 7) externally in the ratio 5 : 3.

StandardTry it yourself first

In what ratio does the X-axis divide the segment joining A(2, -6) and B(5, 9)?

StandardTry it yourself first

In what ratio does the X-axis divide the segment joining A(-3, 4) and B(1, -12)?

AdvancedTry it yourself first

In what ratio does the X-axis divide the segment joining A(7, -8) and B(-1, 2)?

AdvancedTry it yourself first

In what ratio does the X-axis divide the segment joining A(0, -10) and B(4, 15)?

BasicTry it yourself first

The vertices of a triangle are (1, 2), (4, 6) and (7, 1). Find its centroid.

StandardTry it yourself first

The vertices of a triangle are (-2, 5), (3, -1) and (8, 2). Find its centroid.

AdvancedTry it yourself first

The vertices of a triangle are (-5, 4), (9, -6) and (2, 8). Find its centroid.

StandardTry it yourself first

Two vertices of a triangle are (1, 2) and (5, 4) and its centroid is (4, 5). Find the third vertex.

AdvancedTry it yourself first

Two vertices of a triangle are (-3, 1) and (7, 6) and its centroid is (2, 4). Find the third vertex.

StandardTry it yourself first

The mid-points of the sides of a triangle are (2, 3), (5, 6) and (8, 9). Find the centroid of the triangle.

AdvancedTry it yourself first

The mid-points of the sides of a triangle are (1, 4), (3, 7) and (8, 1). Find the centroid of the triangle.

AdvancedTry it yourself first

The mid-points of the sides of a triangle are (-1, 2), (5, 6) and (2, -2). Find the centroid of the triangle.

Area and shapes

Recognising a figure from its coordinates

The area of a triangle and quadrilateral, collinearity, the fourth vertex of a parallelogram and the kind of triangle.

Area, collinearity and identifying a figure

Never forget the modulus

The area formula, the shoelace method, the condition for three collinear points and identifying a figure from the squares of its sides.

∑ Formula

Δ = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

Area of a triangle

Collinear points: the “triangle” has area $0$ ; or $A B$ and $BC$ have equal slopes; or $A B + BC = A C$ .
Area of a quadrilateral: take the vertices in order and use the shoelace sum $\frac{1}{2} ∣ \sum (x_{i} y_{i + 1} - x_{i + 1} y_{i}) ∣$ , closing back to the first vertex.
Identify the figure: find the squares of the three sides — two equal ⟹ isosceles; the largest = sum of the other two ⟹ right-angled; all equal ⟹ equilateral.
Parallelogram ABCD: $D = A + C - B$ . Rectangle/rhombus are identified by diagonals: equal in a rectangle, perpendicular in a rhombus.

Example 6 · Standard

Find the area of the triangle with vertices

(2, 1)

(8, 1)

and

(5, 7)

Solution, step by step

1By the formula: $\frac{1}{2} ∣2 (1 - 7) + 8 (7 - 1) + 5 (1 - 1) ∣ = \frac{1}{2} ∣ - 12 + 48 + 0∣ = 18$ .
2Check: base $= 6$ (both have $y = 1$ ), height $= 7 - 1 = 6$ , area $= \frac{1}{2} \times 6 \times 6 = 18$ ✓

Answer 18 sq units

BasicTry it yourself first

Find the area of the triangle with vertices (0, 0), (6, 0) and (0, 8).

BasicTry it yourself first

Find the area of the triangle with vertices (1, 1), (5, 1) and (3, 6).

StandardTry it yourself first

Find the area of the triangle with vertices (2, 3), (7, 5) and (4, 9).

StandardTry it yourself first

Find the area of the triangle with vertices (-3, 2), (5, -1) and (1, 6).

AdvancedTry it yourself first

Find the area of the triangle with vertices (-4, -2), (6, 3) and (2, 9).

AdvancedTry it yourself first

Find the area of the triangle with vertices (7, -5), (-2, 4) and (3, 8).

StandardTry it yourself first

The vertices of quadrilateral ABCD in order are A(0, 0), B(6, 0), C(6, 4) and D(0, 4). Find its area.

AdvancedTry it yourself first

The vertices of quadrilateral ABCD in order are A(1, 1), B(7, 2), C(8, 6) and D(2, 5). Find its area.

AdvancedTry it yourself first

The vertices of quadrilateral ABCD in order are A(-2, 0), B(3, -3), C(6, 2) and D(1, 5). Find its area.

StandardTry it yourself first

The points (1, 2),

(3, k)

and (5, 10) are collinear. Find

k

StandardTry it yourself first

The points (-2, 1),

(1, k)

and (4, 7) are collinear. Find

k

AdvancedTry it yourself first

The points (2, -3),

(5, k)

and (8, 9) are collinear. Find

k

AdvancedTry it yourself first

The points (-4, 7),

(0, k)

and (6, -3) are collinear. Find

k

StandardTry it yourself first

Three vertices of parallelogram ABCD are A(1, 1), B(4, 2) and C(5, 5). Find the fourth vertex D.

StandardTry it yourself first

Three vertices of parallelogram ABCD are A(-1, 2), B(3, 0) and C(6, 4). Find the fourth vertex D.

AdvancedTry it yourself first

Three vertices of parallelogram ABCD are A(0, 0), B(5, 1) and C(7, 6). Find the fourth vertex D.

AdvancedTry it yourself first

Three vertices of parallelogram ABCD are A(-3, -2), B(2, -4) and C(5, 3). Find the fourth vertex D.

StandardTry it yourself first

What do the points (0, 0), (4, 0) and (0, 3) form?

StandardTry it yourself first

What do the points (1, 1), (5, 1) and (3, 5) form?

StandardTry it yourself first

What do the points (0, 0), (2, 2) and (4, 4) form?

AdvancedTry it yourself first

What do the points (1, 2), (5, 6) and (1, 6) form?

AdvancedTry it yourself first

What do the points (0, 0), (3, 4) and (6, 0) form?

AdvancedTry it yourself first

What do the points (0, 0), (4, 1) and (2, 6) form?

AdvancedTry it yourself first

What do the points (0, 0), (5, 0) and (0, 12) form?

The straight line

Slope and equation

Slope, the forms of a line’s equation, intercepts, parallel and perpendicular lines, distance from a point and points of intersection.

Slope and equation

m = tan θ

Slope, angle of inclination and the five forms of a line’s equation.

◭ Diagram

Slope

m = \frac{rise}{run} = tan θ

Form	Equation
Slope–intercept	$y = m x + c$
Point–slope	$y - y_{1} = m (x - x_{1})$
Two-point	$\frac{y - y _{1}}{y _{2} - y _{1}} = \frac{x - x _{1}}{x _{2} - x _{1}}$
Intercept	$\frac{x}{a} + \frac{y}{b} = 1$
General	$a x + b y + c = 0$ , ढलान $- \frac{a}{b}$

Slope $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = tan θ$ . The X-axis has slope $0$ ; the Y-axis is undefined; $4 5^{\circ}$ gives $1$ ; $13 5^{\circ}$ gives $- 1$ .
A line parallel to the X-axis: $y =$ constant; parallel to the Y-axis: $x =$ constant.
A line with equal intercepts $a$ : $x + y = a$ — put in the given point to find $a$ .

Example 7 · Standard

Find the equation and slope of the line through

(2, 3)

and

(- 1, - 3)

Solution, step by step

1Slope $m = \frac{- 3 - 3}{- 1 - 2} = \frac{- 6}{- 3} = 2$ .
2 $y - 3 = 2 (x - 2)$ ⟹ $2 x - y - 1 = 0$

Answer slope 2; $2 x - y - 1 = 0$

BasicTry it yourself first

Find the slope of the line through (1, 2) and (4, 8).

BasicTry it yourself first

Find the slope of the line through (0, 0) and (4, -12).

StandardTry it yourself first

Find the slope of the line through (2, -1) and (6, 5).

StandardTry it yourself first

Find the slope of the line through (-3, 4) and (5, -2).

AdvancedTry it yourself first

Find the slope of the line through (-5, 7) and (3, -9).

AdvancedTry it yourself first

Find the slope of the line through (4, 11) and (-2, 2).

BasicTry it yourself first

Find the slope of the line

3 x + 4 y - 12 = 0

StandardTry it yourself first

Find the slope of the line

2 x - 5 y + 10 = 0

StandardTry it yourself first

Find the slope of the line

6 x + y - 7 = 0

AdvancedTry it yourself first

Find the slope of the line

4 x - 3 y + 1 = 0

AdvancedTry it yourself first

Find the slope of the line

7 x + 2 y - 9 = 0

StandardTry it yourself first

The line joining

A (x, 3)

and

B (7, 9)

is inclined at

4 5^{\circ}

to the X-axis. Find

x

StandardTry it yourself first

The line joining

A (x, 2)

and

B (4, 10)

is inclined at

13 5^{\circ}

to the X-axis. Find

x

AdvancedTry it yourself first

The line joining

A (x, - 9)

and

B (- 3, 5)

is inclined at

4 5^{\circ}

to the X-axis. Find

x

AdvancedTry it yourself first

The line joining

A (x, 4)

and

B (6, - 2)

is inclined at

13 5^{\circ}

to the X-axis. Find

x

BasicTry it yourself first

Find the equation of the line with slope 2 and Y-intercept -3.

StandardTry it yourself first

Find the equation of the line with slope -3 and Y-intercept 4.

StandardTry it yourself first

Find the equation of the line through (1, 2) with slope 3.

AdvancedTry it yourself first

Find the equation of the line through (3, -4) with slope -2.

StandardTry it yourself first

Find the equation of the line through (1, 2) and (3, 8).

AdvancedTry it yourself first

Find the equation of the line through (-2, 3) and (4, -1).

AdvancedTry it yourself first

Find the equation of the line through (0, 5) and (6, 2).

StandardTry it yourself first

Find the equation of the line cutting intercepts 3 on the X-axis and 4 on the Y-axis.

AdvancedTry it yourself first

Find the equation of the line cutting intercepts -2 on the X-axis and 5 on the Y-axis.

StandardTry it yourself first

Find the equation of the line making equal intercepts on the axes and passing through (2, 7).

AdvancedTry it yourself first

Find the equation of the line making equal intercepts on the axes and passing through (9, -3).

StandardTry it yourself first

Find the equation of the line through (-3, 4) parallel to the Y-axis.

StandardTry it yourself first

Find the equation of the line through (6, -2) parallel to the X-axis.

AdvancedTry it yourself first

Find the equation of the line through (5, 8) parallel to the X-axis.

Intercepts, parallel and perpendicular lines, distance

How two lines relate

Parts cut off by the axes, conditions for parallel and perpendicular lines, point–line distance, intersection and concurrent lines.

Matter	Condition / formula
Parallel lines	$m_{1} = m_{2}$ ; that is $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}}$
Perpendicular lines	$m_{1} m_{2} = - 1$ ; that is $a_{1} a_{2} + b_{1} b_{2} = 0$
Distance of $(x_{1}, y_{1})$ from $a x + b y + c = 0$	$\frac{∣ a x _{1} + b y _{1} + c ∣}{a ^{2} + b ^{2}}$
Distance between parallels	$\frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$
Three lines concurrent	the meeting point of two lies on the third

Intercepts: put $y = 0$ for the X-intercept, $x = 0$ for the Y-intercept. The triangle with the axes has area $\frac{1}{2} ∣ a ∣∣ b ∣$ .
Parallel line: same $a, b$ , new $c$ — find $c$ from the given point. Perpendicular line: replace $a, b$ with $b, - a$ .
Right-angle condition: if $∠ B = 9 0^{\circ}$ , the slopes of $B A$ and $BC$ multiply to $- 1$ .

Example 8 · Standard

Find the area of the triangle bounded by the line

2 x + 3 y = 18

and the two coordinate axes.

Solution, step by step

1 $y = 0$ : $x = 9$ ; $x = 0$ : $y = 6$ .
2Area $= \frac{1}{2} \times 9 \times 6 = 27$ sq units

Answer 27 sq units

Example 9 · Standard

Find the distance of the point

(1, - 2)

from the line

3 x + 4 y - 10 = 0

Solution, step by step

1 $∣3 (1) + 4 (- 2) - 10∣ = ∣3 - 8 - 10∣ = 15$ .
2 $3^{2} + 4^{2} = 5$ ; distance $= \frac{15}{5} = 3$ units

Answer 3 units

BasicTry it yourself first

Find the X-intercept of the line

3 x + 7 y = 21

BasicTry it yourself first

Find the Y-intercept of the line

4 x + 9 y = 36

StandardTry it yourself first

Find the area of the triangle enclosed by the line

6 x + 4 y = 24

and the two coordinate axes.

StandardTry it yourself first

Find the area of the triangle enclosed by the line

2 x + 5 y = 20

and the two coordinate axes.

StandardTry it yourself first

How long is the part of the line

5 x + 12 y = 60

that lies between the two coordinate axes?

AdvancedTry it yourself first

How long is the part of the line

8 x + 15 y = 120

that lies between the two coordinate axes?

AdvancedTry it yourself first

How long is the part of the line

12 x + 5 y = 60

that lies between the two coordinate axes?

AdvancedTry it yourself first

Find the area of the triangle enclosed by the line

5 x + 3 y = 30

and the two coordinate axes.

StandardTry it yourself first

The lines

2 x + 3 y = 6

and

k x + 9 y = 4

are parallel. Find

k

StandardTry it yourself first

The lines

4 x + 5 y = 20

and

k x + 10 y = 3

are parallel. Find

k

AdvancedTry it yourself first

The lines

3 x + 4 y = 12

and

k x + 8 y = 5

are parallel. Find

k

StandardTry it yourself first

The lines

3 x + 4 y = 5

and

8 x + k y = 1

are perpendicular. Find

k

StandardTry it yourself first

The lines

2 x + 5 y = 1

and

10 x + k y = 3

are perpendicular. Find

k

AdvancedTry it yourself first

The lines

4 x + 3 y = 2

and

9 x + k y = 7

are perpendicular. Find

k

StandardTry it yourself first

Find the equation of the line through (4, 1) parallel to

2 x + 3 y - 6 = 0

AdvancedTry it yourself first

Find the equation of the line through (3, 4) parallel to

5 x - 2 y + 7 = 0

StandardTry it yourself first

Find the equation of the line through (1, 4) perpendicular to

3 x + 2 y + 5 = 0

AdvancedTry it yourself first

Find the equation of the line through (2, 5) perpendicular to

4 x - 3 y + 1 = 0

StandardTry it yourself first

Find the slope of a line perpendicular to the line through (1, 2) and (5, 10).

AdvancedTry it yourself first

Find the slope of a line perpendicular to the line through (2, 7) and (9, 1).

StandardTry it yourself first

Find the distance of the point (2, 1) from the line

3 x + 4 y + 5 = 0

StandardTry it yourself first

Find the distance of the point (5, 2) from the line

4 x + 3 y - 7 = 0

AdvancedTry it yourself first

Find the distance of the point (-1, 4) from the line

6 x + 8 y + 3 = 0

AdvancedTry it yourself first

Find the distance of the point (3, -2) from the line

8 x + 6 y - 11 = 0

AdvancedTry it yourself first

Find the distance of the point (0, 0) from the line

3 x + 4 y - 20 = 0

StandardTry it yourself first

Find the distance between the parallel lines

3 x + 4 y + 5 = 0

and

3 x + 4 y - 10 = 0

StandardTry it yourself first

Find the distance between the parallel lines

4 x + 3 y + 6 = 0

and

4 x + 3 y - 4 = 0

AdvancedTry it yourself first

Find the distance between the parallel lines

6 x + 8 y + 12 = 0

and

6 x + 8 y - 18 = 0

AdvancedTry it yourself first

Find the distance between the parallel lines

3 x + 4 y + 1 = 0

and

3 x + 4 y - 9 = 0

StandardTry it yourself first

Find the point where the lines

x + y = 7

and

2 x - y = 2

meet.

StandardTry it yourself first

Find the point where the lines

2 x + 3 y = 13

and

x - 2 y = - 4

meet.

AdvancedTry it yourself first

Find the point where the lines

4 x + y = 14

and

x + 2 y = 7

meet.

AdvancedTry it yourself first

Find the point where the lines

3 x - 2 y = 4

and

5 x + 2 y = 28

meet.

StandardTry it yourself first

The lines

x + y = 7

2 x - y = 2

and

5 x + 2 y = k

pass through one point. Find

k

AdvancedTry it yourself first

The lines

2 x + 3 y = 13

x - 2 y = - 4

and

4 x - y = k

pass through one point. Find

k

AdvancedTry it yourself first

The lines

4 x + y = 14

x + 2 y = 7

and

3 x + 5 y = k

pass through one point. Find

k

StandardTry it yourself first

The points A(2, 5), B(4, 1) and

C (10, k)

are such that

∠ A BC = 9 0^{\circ}

. Find

k

AdvancedTry it yourself first

The points A(0, 6), B(3, 2) and

C (11, k)

are such that

∠ A BC = 9 0^{\circ}

. Find

k

AdvancedTry it yourself first

The points A(-1, 1), B(2, -3) and

C (6, k)

are such that

∠ A BC = 9 0^{\circ}

. Find

k

Traps and fast tricks

Slips · tricks

Where marks go in coordinate geometry.

The commonest mistakes

Seven traps

The slips that cost marks.

Trap 1Sign slips with negative numbers. If

x_{1} = - 3

then

x_{2} - x_{1} = x_{2} - (- 3) = x_{2} + 3

— subtract with brackets.

Trap 2Adding instead of subtracting in distance.

(x_{2} + x_{1})^{2} + \dots

is wrong; always square the difference.

Trap 3Forgetting the modulus and the $\frac{1}{2}$ in area. The formula can come out negative — take

∣ ∣

, then half.

Trap 4Swapping $m$ and $n$ in the section formula.

m

goes with the coordinates of the second end

B

n

with the first end

A

Trap 5The sign of a perpendicular slope. The slope of a perpendicular is

- \frac{1}{m}

— flip and change the sign; flipping alone is not enough.

Trap 6Zeroing the wrong variable for an intercept. Put

y = 0

for the X-intercept;

x = 0

for the Y-intercept.

Trap 7Treating $a^{2} + b^{2}$ as $a + b$ in the distance formula, and forgetting the modulus in

∣ a x_{1} + b y_{1} + c ∣

Fast tricks

Recognise and apply

Habits for speed.

Sketch a small figure: placing the points on a graph often shows the answer and avoids sign slips.
Check: an internal division point lies between the ends, an external one lies outside; for the mid-point $m = n$ .
Test for collinearity: area $0$ ⟺ equal slopes ⟺ distances add up.
After writing a line’s equation, put the given points back in to check it.
The Pythagorean triples ( $3 - 4 - 5$ , $5 - 12 - 13$ , $8 - 15 - 17$ ) give distances and intercept lengths at once.

Example 10 · Standard

Are the points

A (1, 2)

B (4, 6)

and

C (7, 10)

collinear? Check in two ways.

Solution, step by step

1Slopes: $A B = \frac{6 - 2}{4 - 1} = \frac{4}{3}$ and $BC = \frac{10 - 6}{7 - 4} = \frac{4}{3}$ — equal.
2Distances: $A B = 5$ , $BC = 5$ , $A C = 10$ and $5 + 5 = 10$ ✓ — collinear

Answer Yes, collinear

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct formulas

One-formula questions.

BasicTry it yourself first

Where does the point (5, 3) lie?

BasicTry it yourself first

Where does the point (-2, -6) lie?

BasicTry it yourself first

Where does the point (-9, 4) lie?

BasicTry it yourself first

Where does the point (8, -1) lie?

BasicTry it yourself first

Where does the point (0, 3) lie?

BasicTry it yourself first

Find the reflection of the point (-2, 6) in the X-axis.

BasicTry it yourself first

Find the reflection of the point (5, -3) in the Y-axis.

BasicTry it yourself first

Find the reflection of the point (-4, -1) in the origin.

BasicTry it yourself first

Find the distance between the points (3, 1) and (3, 7).

BasicTry it yourself first

Find the distance between the points (2, 2) and (14, 7).

BasicTry it yourself first

Find the distance of the point (3, 4) from the origin.

BasicTry it yourself first

Find the distance of the point (9, 12) from the origin.

BasicTry it yourself first

Find the mid-point of the segment joining (0, 0) and (6, 10).

BasicTry it yourself first

Find the mid-point of the segment joining (-2, -2) and (6, 8).

BasicTry it yourself first

The mid-point of segment AB is (1, 1) and A is (3, -1). Find the coordinates of B.

BasicTry it yourself first

Find the point that divides the segment joining A(0, 0) and B(9, 6) internally in the ratio 1 : 2.

BasicTry it yourself first

In what ratio does the X-axis divide the segment joining A(1, -4) and B(3, 8)?

BasicTry it yourself first

The vertices of a triangle are (0, 0), (6, 3) and (3, 9). Find its centroid.

BasicTry it yourself first

Find the area of the triangle with vertices (0, 0), (4, 0) and (0, 6).

BasicTry it yourself first

Find the area of the triangle with vertices (1, 2), (7, 2) and (4, 8).

BasicTry it yourself first

Find the slope of the line through (0, 1) and (4, 9).

BasicTry it yourself first

Find the slope of the line through (1, 5) and (3, 1).

BasicTry it yourself first

Find the slope of the line

2 x + y - 4 = 0

BasicTry it yourself first

Find the equation of the line with slope 1 and Y-intercept 5.

BasicTry it yourself first

Find the equation of the line through (4, 9) parallel to the Y-axis.

BasicTry it yourself first

Find the X-intercept of the line

2 x + 5 y = 10

BasicTry it yourself first

Find the Y-intercept of the line

3 x + 4 y = 24

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

Where does the point (-7, -1) lie?

StandardTry it yourself first

Where does the point (3, -9) lie?

StandardTry it yourself first

Where does the point (0, -4) lie?

StandardTry it yourself first

Where does the point (-6, 0) lie?

StandardTry it yourself first

Find the reflection of the point (7, 2) in the X-axis.

StandardTry it yourself first

Find the reflection of the point (-3, -8) in the Y-axis.

StandardTry it yourself first

Find the reflection of the point (5, -6) in the origin.

StandardTry it yourself first

Find the reflection of the point (-2, 9) in the line

y = x

StandardTry it yourself first

Find the distance between the points (-3, -4) and (2, 8).

StandardTry it yourself first

Find the distance between the points (6, -2) and (-2, 4).

StandardTry it yourself first

Find the distance between the points (2, 3) and (7, 9).

StandardTry it yourself first

Find the distance between the points (-5, 1) and (1, 5).

StandardTry it yourself first

Find the distance of the point (-9, 12) from the origin.

StandardTry it yourself first

Find the distance of the point (7, -1) from the origin.

StandardTry it yourself first

Find the distance of the point (10, 24) from the origin.

StandardTry it yourself first

The distance between the points

(x, 4)

and (-2, 1) is 5 units. Find the possible values of

x

StandardTry it yourself first

The distance between the points

(x, 9)

and (4, -3) is 15 units. Find the possible values of

x

StandardTry it yourself first

The distance between the points

(x, - 7)

and (0, 5) is 13 units. Find the possible values of

x

StandardTry it yourself first

The distance between the points

(x, - 8)

and (6, 0) is 10 units. Find the possible values of

x

StandardTry it yourself first

Find the equation of the locus of points equidistant from (2, -1) and (6, 3).

StandardTry it yourself first

Find the equation of the locus of points equidistant from (-1, 0) and (3, 6).

StandardTry it yourself first

Find the point on the X-axis equidistant from (2, 1) and (6, 3).

StandardTry it yourself first

Find the point on the X-axis equidistant from (0, 4) and (6, 2).

StandardTry it yourself first

Find the point on the X-axis equidistant from (3, 2) and (9, 4).

StandardTry it yourself first

Find the point on the X-axis equidistant from (-2, 5) and (4, 1).

StandardTry it yourself first

Find the mid-point of the segment joining (7, -4) and (-3, 10).

StandardTry it yourself first

The mid-point of segment AB is (4, -2) and A is (9, 5). Find the coordinates of B.

StandardTry it yourself first

Find the point that divides the segment joining A(3, -1) and B(13, 9) internally in the ratio 2 : 3.

StandardTry it yourself first

Find the point that divides the segment joining A(1, 1) and B(4, 7) externally in the ratio 3 : 1.

StandardTry it yourself first

In what ratio does the X-axis divide the segment joining A(6, -5) and B(-2, 10)?

StandardTry it yourself first

In what ratio does the X-axis divide the segment joining A(3, -15) and B(9, 5)?

StandardTry it yourself first

The vertices of a triangle are (4, -2), (-7, 9) and (0, 5). Find its centroid.

StandardTry it yourself first

Two vertices of a triangle are (2, -5) and (8, 1) and its centroid is (3, -1). Find the third vertex.

StandardTry it yourself first

The mid-points of the sides of a triangle are (6, 2), (0, 8) and (9, 5). Find the centroid of the triangle.

StandardTry it yourself first

Find the area of the triangle with vertices (0, -3), (8, 1) and (2, 6).

StandardTry it yourself first

Find the area of the triangle with vertices (-2, -2), (4, 1) and (1, 7).

StandardTry it yourself first

The vertices of quadrilateral ABCD in order are A(0, 0), B(5, -2), C(8, 3) and D(1, 6). Find its area.

StandardTry it yourself first

The points (3, -1),

(7, k)

and (11, 5) are collinear. Find

k

StandardTry it yourself first

The points (-3, 8),

(0, k)

and (3, -4) are collinear. Find

k

StandardTry it yourself first

Three vertices of parallelogram ABCD are A(2, 3), B(6, 1) and C(9, 7). Find the fourth vertex D.

StandardTry it yourself first

What do the points (1, 1), (7, 1) and (4, 5) form?

StandardTry it yourself first

What do the points (1, 1), (1, 5) and (4, 1) form?

StandardTry it yourself first

What do the points (2, 3), (5, 7) and (9, 4) form?

StandardTry it yourself first

Find the slope of the line through (3, -2) and (7, 4).

StandardTry it yourself first

Find the slope of the line through (-1, 6) and (5, -3).

StandardTry it yourself first

Find the slope of the line

5 x - 2 y + 3 = 0

StandardTry it yourself first

Find the slope of the line

3 x + 7 y - 1 = 0

StandardTry it yourself first

The line joining

A (x, 5)

and

B (2, 11)

is inclined at

4 5^{\circ}

to the X-axis. Find

x

StandardTry it yourself first

The line joining

A (x, - 2)

and

B (- 1, 8)

is inclined at

13 5^{\circ}

to the X-axis. Find

x

StandardTry it yourself first

Find the equation of the line with slope 4 and Y-intercept -1.

StandardTry it yourself first

Find the equation of the line through (-1, 3) with slope 2.

StandardTry it yourself first

Find the equation of the line through (2, 1) and (5, 7).

StandardTry it yourself first

Find the equation of the line cutting intercepts 5 on the X-axis and 2 on the Y-axis.

StandardTry it yourself first

Find the equation of the line making equal intercepts on the axes and passing through (3, 5).

StandardTry it yourself first

Find the equation of the line through (1, -6) parallel to the X-axis.

StandardTry it yourself first

Find the X-intercept of the line

5 x + 6 y = 30

StandardTry it yourself first

Find the Y-intercept of the line

7 x + 2 y = 28

StandardTry it yourself first

Find the area of the triangle enclosed by the line

4 x + 3 y = 24

and the two coordinate axes.

StandardTry it yourself first

How long is the part of the line

9 x + 12 y = 36

that lies between the two coordinate axes?

StandardTry it yourself first

The lines

3 x + 5 y = 15

and

k x + 10 y = 6

are parallel. Find

k

StandardTry it yourself first

The lines

5 x + 2 y = 3

and

4 x + k y = 0

are perpendicular. Find

k

StandardTry it yourself first

Find the equation of the line through (5, 2) parallel to

3 x - 4 y + 2 = 0

StandardTry it yourself first

Find the equation of the line through (4, -3) perpendicular to

2 x + 5 y + 1 = 0

StandardTry it yourself first

Find the slope of a line perpendicular to the line through (3, 1) and (7, 9).

StandardTry it yourself first

Find the distance of the point (3, -3) from the line

4 x + 3 y - 1 = 0

StandardTry it yourself first

Find the distance of the point (2, 3) from the line

6 x + 8 y - 5 = 0

StandardTry it yourself first

Find the distance between the parallel lines

4 x + 3 y + 10 = 0

and

4 x + 3 y - 5 = 0

StandardTry it yourself first

Find the distance between the parallel lines

3 x + 4 y - 7 = 0

and

3 x + 4 y + 3 = 0

StandardTry it yourself first

Find the distance between the parallel lines

6 x + 8 y + 9 = 0

and

6 x + 8 y - 1 = 0

StandardTry it yourself first

Find the point where the lines

2 x + y = 11

and

x - y = 1

meet.

StandardTry it yourself first

Find the point where the lines

3 x + 4 y = 24

and

x + 2 y = 10

meet.

StandardTry it yourself first

The lines

2 x + y = 11

x - y = 1

and

3 x + 2 y = k

pass through one point. Find

k

StandardTry it yourself first

The points A(3, 7), B(5, 1) and

C (11, k)

are such that

∠ A BC = 9 0^{\circ}

. Find

k

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

Where does the point (-15, -4) lie?

AdvancedTry it yourself first

Where does the point (0, -11) lie?

AdvancedTry it yourself first

Find the reflection of the point (-9, -7) in the X-axis.

AdvancedTry it yourself first

Find the reflection of the point (12, -5) in the line

y = x

AdvancedTry it yourself first

Find the reflection of the point (-8, 3) in the origin.

AdvancedTry it yourself first

Find the distance between the points (-7, -3) and (5, 2).

AdvancedTry it yourself first

Find the distance between the points (9, -4) and (-3, 5).

AdvancedTry it yourself first

Find the distance between the points (-2, 6) and (8, -3).

AdvancedTry it yourself first

Find the distance between the points (11, 2) and (-5, -10).

AdvancedTry it yourself first

Find the distance of the point (-20, 21) from the origin.

AdvancedTry it yourself first

Find the distance of the point (11, -7) from the origin.

AdvancedTry it yourself first

The distance between the points

(x, 11)

and (7, -1) is 15 units. Find the possible values of

x

AdvancedTry it yourself first

The distance between the points

(x, - 4)

and (-5, 4) is 17 units. Find the possible values of

x

AdvancedTry it yourself first

Find the equation of the locus of points equidistant from (-3, 2) and (5, -4).

AdvancedTry it yourself first

Find the equation of the locus of points equidistant from (6, 1) and (-2, -5).

AdvancedTry it yourself first

Find the point on the X-axis equidistant from (2, 5) and (8, 1).

AdvancedTry it yourself first

Find the point on the X-axis equidistant from (-3, 2) and (5, 6).

AdvancedTry it yourself first

Find the mid-point of the segment joining (-13, 8) and (9, -6).

AdvancedTry it yourself first

The mid-point of segment AB is (-5, 3) and A is (-1, -7). Find the coordinates of B.

AdvancedTry it yourself first

Find the point that divides the segment joining A(-4, 2) and B(11, 17) internally in the ratio 4 : 1.

AdvancedTry it yourself first

Find the point that divides the segment joining A(2, -3) and B(10, 5) externally in the ratio 7 : 3.

AdvancedTry it yourself first

In what ratio does the X-axis divide the segment joining A(-2, -9) and B(6, 12)?

AdvancedTry it yourself first

In what ratio does the X-axis divide the segment joining A(1, 14) and B(9, -6)?

AdvancedTry it yourself first

The vertices of a triangle are (-8, 6), (5, -9) and (12, 3). Find its centroid.

AdvancedTry it yourself first

Two vertices of a triangle are (-4, 7) and (6, -2) and its centroid is (1, 3). Find the third vertex.

AdvancedTry it yourself first

The mid-points of the sides of a triangle are (-3, 4), (5, -1) and (7, 6). Find the centroid of the triangle.

AdvancedTry it yourself first

Find the area of the triangle with vertices (-5, -3), (7, 2) and (-1, 9).

AdvancedTry it yourself first

Find the area of the triangle with vertices (6, -4), (-3, 5) and (0, -8).

AdvancedTry it yourself first

The vertices of quadrilateral ABCD in order are A(-3, -2), B(4, -5), C(9, 2) and D(1, 8). Find its area.

AdvancedTry it yourself first

The points (-2, -9),

(1, k)

and (10, 15) are collinear. Find

k

AdvancedTry it yourself first

The points (5, -2),

(- 1, k)

and (11, 10) are collinear. Find

k

AdvancedTry it yourself first

Three vertices of parallelogram ABCD are A(-4, 0), B(1, 5) and C(8, 2). Find the fourth vertex D.

AdvancedTry it yourself first

What do the points (-2, 1), (2, 4) and (5, 0) form?

AdvancedTry it yourself first

What do the points (0, 0), (8, 6) and (16, 0) form?

AdvancedTry it yourself first

What do the points (1, 1), (4, 5) and (10, -7) form?

AdvancedTry it yourself first

Find the slope of the line through (-6, -2) and (4, 8).

AdvancedTry it yourself first

Find the slope of the line through (9, -5) and (-3, 7).

AdvancedTry it yourself first

Find the slope of the line

9 x - 6 y + 2 = 0

AdvancedTry it yourself first

Find the slope of the line

- 4 x + 5 y + 1 = 0

AdvancedTry it yourself first

The line joining

A (x, - 4)

and

B (- 5, 3)

is inclined at

4 5^{\circ}

to the X-axis. Find

x

AdvancedTry it yourself first

The line joining

A (x, 2)

and

B (3, - 6)

is inclined at

13 5^{\circ}

to the X-axis. Find

x

AdvancedTry it yourself first

Find the equation of the line with slope -5 and Y-intercept -2.

AdvancedTry it yourself first

Find the equation of the line through (-3, 5) with slope 4.

AdvancedTry it yourself first

Find the equation of the line through (-3, -2) and (5, 4).

AdvancedTry it yourself first

Find the equation of the line cutting intercepts -6 on the X-axis and 4 on the Y-axis.

AdvancedTry it yourself first

Find the equation of the line making equal intercepts on the axes and passing through (-4, 10).

AdvancedTry it yourself first

Find the equation of the line through (-7, -5) parallel to the Y-axis.

AdvancedTry it yourself first

Find the X-intercept of the line

9 x + 4 y = 72

AdvancedTry it yourself first

Find the Y-intercept of the line

7 x + 6 y = 84

AdvancedTry it yourself first

Find the area of the triangle enclosed by the line

3 x + 8 y = 48

and the two coordinate axes.

AdvancedTry it yourself first

How long is the part of the line

7 x + 24 y = 168

that lies between the two coordinate axes?

AdvancedTry it yourself first

The lines

6 x + 5 y = 30

and

k x + 15 y = 4

are parallel. Find

k

AdvancedTry it yourself first

The lines

7 x + 3 y = 2

and

6 x + k y = 0

are perpendicular. Find

k

AdvancedTry it yourself first

Find the equation of the line through (2, -5) parallel to

7 x - 3 y + 1 = 0

AdvancedTry it yourself first

Find the equation of the line through (-1, 6) perpendicular to

5 x - 2 y + 3 = 0

AdvancedTry it yourself first

Find the slope of a line perpendicular to the line through (-2, 3) and (4, 11).

AdvancedTry it yourself first

Find the distance of the point (4, -1) from the line

8 x + 6 y - 9 = 0

AdvancedTry it yourself first

Find the distance of the point (-2, 6) from the line

3 x + 4 y - 7 = 0

AdvancedTry it yourself first

Find the distance of the point (-5, 1) from the line

4 x + 3 y + 2 = 0

AdvancedTry it yourself first

Find the distance between the parallel lines

8 x + 6 y + 15 = 0

and

8 x + 6 y - 5 = 0

AdvancedTry it yourself first

Find the distance between the parallel lines

4 x + 3 y - 12 = 0

and

4 x + 3 y + 8 = 0

AdvancedTry it yourself first

Find the distance between the parallel lines

6 x + 8 y - 7 = 0

and

6 x + 8 y + 13 = 0

AdvancedTry it yourself first

Find the point where the lines

5 x + 2 y = 16

and

3 x - 2 y = 0

meet.

AdvancedTry it yourself first

Find the point where the lines

3 x + y = 1

and

x + 2 y = 7

meet.

AdvancedTry it yourself first

The lines

5 x + 2 y = 16

3 x - 2 y = 0

and

7 x - y = k

pass through one point. Find

k

AdvancedTry it yourself first

The lines

3 x + y = 1

x + 2 y = 7

and

2 x + 5 y = k

pass through one point. Find

k

AdvancedTry it yourself first

The points A(-2, 4), B(1, -2) and

C (7, k)

are such that

∠ A BC = 9 0^{\circ}

. Find

k

AdvancedTry it yourself first

The points A(5, -1), B(2, 3) and

C (- 6, k)

are such that

∠ A BC = 9 0^{\circ}

. Find

k

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.