A-Level Maths | Co-ordinate Geometry

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 $(-7, -2)$

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a) $2$   b) $\tfrac{1}{3}$   c) $-1$   d) $\tfrac{3}{2}$   e) $-\tfrac{2}{3}$   f) $\tfrac{4}{3}$   g) $-4$   h) $\tfrac{8}{15}$

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 $\left(-1, -\tfrac{3}{5}\right)$

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a) $(4, 3)$   b) $(4, 7)$   c) $(-1, -3)$   d) $(1, -6)$   e) $\left(\tfrac{3}{2}, \tfrac{9}{2}\right)$   f) $\left(\tfrac{3}{2}, -\tfrac{7}{2}\right)$

g) $(1.5, 3.8)$   h) $\left(\tfrac{1}{4}, \tfrac{9}{4}\right)$   i) $\left(-\tfrac{9}{8}, \tfrac{7}{10}\right)$

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 $(2, -7)$

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a) $5$   b) $\sqrt{10}$   c) $17$   d) $20$   e) $\sqrt{29}$   f) $4\sqrt{5}$

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

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a) gradient $= -1$, $y$-intercept $= -3$

b) gradient $= \tfrac{1}{2}$, $y$-intercept $= -3$

c) gradient $= -1$, $y$-intercept $= \tfrac{2}{3}$

d) gradient $= \tfrac{4}{5}$, $y$-intercept $= \tfrac{1}{5}$

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) $5x - 3y = 10$

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a) $\left(-\tfrac{5}{2}, 0\right)$ and $(0, 5)$

b) $(-6, 0)$ and $(0, 2)$

c) $\left(0, \tfrac{3}{4}\right)$ and $\left(\tfrac{3}{2}, 0\right)$

d) $\left(0, -\tfrac{10}{3}\right)$ and $(2, 0)$

Question 6

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

a) $m = 1$,   $(2, -4)$

b) $m = \tfrac{1}{2}$,   $(6, 1)$

c) $m = -4$,   $(-1, 8)$

d) $m = \tfrac{2}{5}$,   $(-3, 5)$

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

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

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a) $x - y - 6 = 0$   b) $x - 2y - 4 = 0$   c) $4x + y - 4 = 0$

d) $2x - 5y + 31 = 0$   e) $24x + 8y - 35 = 0$   f) $3x + 4y + 26 = 0$

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 $x + 4y - 2 = 0$

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a) $(2, 5)$   b) $(-1, 6)$   c) $\left(\tfrac{3}{2}, \tfrac{7}{2}\right)$   d) $(0, 2)$   e) $(3, -4)$   f) $\left(-\tfrac{2}{5}, \tfrac{3}{5}\right)$

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

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a) centre $(0,0)$, radius $4$

b) centre $(6,1)$, radius $9$

c) centre $(-1,4)$, radius $11$

d) centre $(7,0)$, radius $0.3$

e) centre $(-2,-5)$, radius $4\sqrt{2}$

f) centre $(8,-9)$, radius $6\sqrt{3}$

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

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a) centre $(0,2)$, radius $1$

b) centre $(1,5)$, radius $7$

c) centre $(-6,4)$, radius $4$

d) centre $(1,-8)$, radius $10$

e) centre $(4,-3)$, radius $5$

f) centre $(-5,1)$, radius $3\sqrt{5}$

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

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a) outside

b) inside

c) on the circle

d) outside