> ## Content Index
> Fetch the complete content index at: https://www.a-level-maths-tutor.com/llms.txt
> Use this file to discover other available public pages before exploring further.

# A-Level Maths | Co-ordinate Geometry
- URL: https://www.a-level-maths-tutor.com/a-level-maths-coordinate-geometry/
- Published: 2026-04-14T12:37:19.000Z
- Updated: 2026-07-25T10:15:25.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

# Co-ordinates

Gradient $\\displaystyle m = \\frac{y\_2 - y\_1}{x\_2 - x\_1}$

**Worked Example**

Find the gradient of the line segment joining $(2,3)$ and $(6,11)$.

$m = \\dfrac{11-3}{6-2} = \\dfrac{8}{4} = 2$

## 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 $(-7, -2)$

Show Answers 

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

Midpoint $\\displaystyle\\left(\\frac{x\_1 + x\_2}{2},\\ \\frac{y\_1 + y\_2}{2}\\right)$

**Worked Example**

Find the midpoint of $(2,4)$ and $(8,10)$.

Midpoint $= \\left(\\dfrac{2+8}{2}, \\dfrac{4+10}{2}\\right) = (5, 7)$

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

Show Answers 

**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)** $\\left(-\\tfrac{9}{8}, \\tfrac{7}{10}\\right)$

Length $\\displaystyle l^2 = (x\_2 - x\_1)^2 + (y\_2 - y\_1)^2$

**Worked Example**

Find the length of the line segment joining $(0,0)$ and $(6,8)$.

$l^2 = 6^2 + 8^2 = 36 + 64 = 100$

$l = 10$

## 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 $(2, -7)$

Show Answers 

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

# 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

**Worked Example**

Find the gradient and $y$-intercept of $2x - 4y + 8 = 0$

$4y = 2x + 8$

$y = \\tfrac{1}{2}x + 2$

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

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

Show Answers 

**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 $= \\tfrac{1}{5}$

**Worked Example**

Find where $2x - y + 4 = 0$ crosses the coordinate axes.

$x = 0: \\; -y+4=0 \\Rightarrow y=4 \\;\\Rightarrow (0,4)$

$y = 0: \\; 2x+4=0 \\Rightarrow x=-2 \\;\\Rightarrow (-2,0)$

## 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)** $5x - 3y = 10$

Show Answers 

**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 $(2, 0)$

**Worked Example**

Find the equation of the line with gradient $3$ through $P(1, 2)$.

$y - 2 = 3(x - 1)$

$y = 3x - 1$

$3x - y - 1 = 0$

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

Show Answers 

**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)** $3x + 4y + 26 = 0$

**Worked Example**

Find where $y = x + 2$ and $y = 3x - 4$ intersect.

$x + 2 = 3x - 4$

$2x = 6 \\Rightarrow x = 3$

$y = 3 + 2 = 5$

Intersect at $(3, 5)$

## 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 $x + 4y - 2 = 0$

Show Answers 

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

# 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

**Worked Example**

Find the centre and radius of $(x-2)^2 + (y+3)^2 = 25$

Centre $= (2, -3)$, radius $= 5$

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

Show Answers 

**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 $6\\sqrt{3}$

**Worked Example**

Find the centre and radius of $x^2 + y^2 - 6x + 4y - 3 = 0$

$(x-3)^2 + (y+2)^2 = 3 + 9 + 4$

$(x-3)^2 + (y+2)^2 = 16$

Centre $= (3, -2)$, radius $= 4$

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

Show Answers 

**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 $3\\sqrt{5}$

**Worked Example**

Is $(1,1)$ inside, outside or on the circle $x^2 + y^2 = 4$?

Centre $(0,0)$, radius$^2 = 4$

distance$^2 = 1^2 + 1^2 = 2$

$2 < 4 \\;\\Rightarrow$ inside the circle

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

Show Answers 

**a)** outside (distance$^2=81$, radius$^2=64$)

**b)** inside (distance$^2=25$, radius$^2=36$)

**c)** on the circle (distance$^2=169=$ radius$^2$)

**d)** outside (distance$^2=34$, radius$^2=30$)

---

### A-Level Maths Tutoring

I offer one-to-one and small group A-Level Maths tutoring for students across the UK and internationally. With 94+ five-star Google reviews and tutoring experience since 2017, I specialise in helping students understand difficult concepts and improve their exam technique.

[A-Level Maths Tutoring](https://www.a-level-maths-tutor.com/a-level-maths-tutor/)

[94+ 5-star Google Reviews](https://www.a-level-maths-tutor.com/reviews/)

[Contact Me](https://www.a-level-maths-tutor.com/contact/)

[Back to A-Level Maths Revision](https://www.a-level-maths-tutor.com/a-level-maths-revision/)