A-Level Maths | Co-ordinate Geometry

Co-ordinates

Gradient — $m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Question 1

Find the gradient of the line segment joining each pair of points.

(a) $(3, 1)$ and $(5, 5)$

(b) $(4, 7)$ and $(10, 9)$

(c) $(6, 1)$ and $(2, 5)$

(d) $(-2, 2)$ and $(2, 8)$

(e) $(1, 3)$ and $(7, -1)$

(f) $(4, 5)$ and $(-5, -7)$

(g) $(-2, 0)$ and $(0, -8)$

(h) $(8, 6)$ and $(-7, -2)$

Midpoint — $\left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)$

Question 2

Find the coordinates of the mid-point of the line segment joining each pair of points.

(a) $(0, 2)$ and $(8, 4)$

(b) $(1, 9)$ and $(7, 5)$

(c) $(-5, 1)$ and $(3, -7)$

(d) $(-5, -7)$ and $(7, -5)$

(e) $(1, 0)$ and $(2, 9)$

(f) $(-1, -2)$ and $(4, -5)$

(g) $(2.4, 3.1)$ and $(0.6, 4.5)$

(h) $(0, 3)$ and $\left(\tfrac{1}{2}, \tfrac{3}{2}\right)$

(i) $\left(-\tfrac{5}{4}, 2\right)$ and $\left(-1, -\tfrac{3}{5}\right)$

Length — $l^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

Question 3

Find the exact length of the line segment joining each pair of points, giving your answers in terms of surds where appropriate.

(a) $(1, 1)$ and $(4, 5)$

(b) $(0, 0)$ and $(3, 1)$

(c) $(1, -4)$ and $(9, 11)$

(d) $(7, -8)$ and $(-9, 4)$

(e) $(3, 12)$ and $(1, 7)$

(f) $(-6, -3)$ and $(2, -7)$

Equation of a Line

Equation of a line — $y = mx + c$   or   $y - y_1 = m(x - x_1)$

Parallel lines — same gradient  ·  Perpendicular lines — gradient is the negative reciprocal

Question 4

Find the gradient and $y$-intercept of each line.

(a) $x + y + 3 = 0$

(b) $x - 2y - 6 = 0$

(c) $3x + 3y - 2 = 0$

(d) $4x - 5y + 1 = 0$

Question 5

Find the coordinates of the points at which each straight line crosses the coordinate axes.

(a) $y = 2x + 5$

(b) $x - 3y + 6 = 0$

(c) $2x + 4y - 3 = 0$

(d) $5x - 3y = 10$

Question 6

Find, in each case, the equation of the straight line with gradient $m$ which passes through the point $P$. Give your answers in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

(a) $m = 1$,   $P(2, -4)$

(b) $m = \tfrac{1}{2}$,   $P(6, 1)$

(c) $m = -4$,   $P(-1, 8)$

(d) $m = \tfrac{2}{5}$,   $P(-3, 5)$

(e) $m = -3$,   $P\!\left(\tfrac{3}{2}, -\tfrac{1}{8}\right)$

(f) $m = -\tfrac{3}{4}$,   $P\!\left(\tfrac{2}{3}, -7\right)$

Question 7

Find the coordinates of the point of intersection of each pair of straight lines.

(a) $y = 2x + 1$ and $y = 3x - 1$

(b) $y = x + 7$ and $y = 4 - 2x$

(c) $y = 5x - 4$ and $y = 3x - 1$

(d) $x + 2y - 4 = 0$ and $3x - 2y + 4 = 0$

(e) $2x + y - 2 = 0$ and $x + 3y + 9 = 0$

(f) $3x + 2y = 0$ and $x + 4y - 2 = 0$

Equation of a Circle

Equation of a circle — $(x - a)^2 + (y - b)^2 = r^2$ where $(a, b)$ is the centre and $r$ is the radius

Complete the square to find the centre and radius when the equation is expanded

Question 8

Write down the coordinates of the centre and the radius of each of the following circles.

(a) $x^2 + y^2 = 16$

(b) $(x - 6)^2 + (y - 1)^2 = 81$

(c) $(x + 1)^2 + (y - 4)^2 = 121$

(d) $(x - 7)^2 + y^2 = 0.09$

(e) $(x + 2)^2 + (y + 5)^2 = 32$

(f) $(x - 8)^2 + (y + 9)^2 = 108$

Question 9

Find the coordinates of the centre and the radius of each of the following circles.

(a) $x^2 + y^2 - 4y + 3 = 0$

(b) $x^2 + y^2 - 2x - 10y - 23 = 0$

(c) $x^2 + y^2 + 12x - 8y + 36 = 0$

(d) $x^2 + y^2 - 2x + 16y = 35$

(e) $x^2 + y^2 = 8x - 6y$

(f) $x^2 + y^2 + 10x - 2y - 19 = 0$

Question 10

Find in each case whether the given point lies inside, outside or on the given circle.

(a) $(0, -9)$    $x^2 + y^2 = 64$

(b) $(4, 7)$    $x^2 + y^2 - 2x - 6y - 26 = 0$

(c) $(7, -3)$    $x^2 + y^2 + 10x - 4y = 140$

(d) $(-4, 1)$    $x^2 + y^2 + 2x + 8y - 13 = 0$