A-Level Maths | Co-ordinate Geometry
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 $(-7, -2)$
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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) $\tfrac{8}{15}$
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 $\left(-1, -\tfrac{3}{5}\right)$
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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) $\left(-\tfrac{9}{8}, \tfrac{7}{10}\right)$
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 $(2, -7)$
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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) $4x - 5y + 1 = 0$
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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 $= \tfrac{1}{5}$
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) $5x - 3y = 10$
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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 $(2, 0)$
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) $m = -\tfrac{3}{4}$, $P\!\left(\tfrac{2}{3}, -7\right)$
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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) $3x + 4y + 26 = 0$
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 $x + 4y - 2 = 0$
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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) $\left(-\tfrac{2}{5}, \tfrac{3}{5}\right)$
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) $(x - 8)^2 + (y + 9)^2 = 108$
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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 $6\sqrt{3}$
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) $x^2 + y^2 + 10x - 2y - 19 = 0$
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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 $3\sqrt{5}$
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) $(-4, 1)$ $x^2 + y^2 + 2x + 8y - 13 = 0$
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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$^2=30$)
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