Coordinate Geometry

Competitive Mathematics · Geometry

6 Part5 worked examples68 practice questions

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.

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.

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.

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.

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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.

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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.

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Traps and fast tricks

Slips · tricks

Where marks go in coordinate geometry.

The commonest mistakes

Seven traps

The slips that cost marks.

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Fast tricks

Recognise and apply

Habits for speed.

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The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct formulas

One-formula questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Coordinate Geometry

Points, distance and division

Axes, quadrants and reflection

Distance and equidistant points

Mid-point, section and centroid

Area and shapes

Area, collinearity and identifying a figure

The straight line

Slope and equation

Intercepts, parallel and perpendicular lines, distance

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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