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iGCSE (2021 Edition)

20.09 Distance from a point to a line (Extended)

Worksheet
Distance from a point to a line
1

Write down the formula to find the perpendicular distance d from the point \left(x_1, y_1\right) to the line A x + B y + C = 0.

2

Find the perpendicular distance from the following points to the given line:

a

Point \left(0,0\right), Line : 4 x + 7 y + 5 = 0

b

Point \left(- 9,-3\right), Line : 6 x + 3 y + 1 = 0

c

Point \left(3, 5\right), Line : 4 x + 3 y + 1 = 0

d

Point \left(4, - 3 \right), Line : 24 x + 7 y - 3 = 0

e

Point \left(5, 4\right), Line : 5 x - 12 y = 0

f

Point \left( - 5 , 3\right), Line : 3 x - 2 y - 5 = 0

3

Find the perpendicular distance from the point (8, 7) to the line 9 x - 9 y - 0 = 0.

4

Find the value of k with the following conditions:

a

The perpendicular distance from the point \left(- 5,-1\right) to the line - 2 x - 7 y + k = 0 is \dfrac{10}{\sqrt{53}}.

b

The perpendicular distance from the point (\left(6,k\right), to the line 6 x + \left( - 2 y\right) + \left( - 6 \right) = 0 is \dfrac{40}{\sqrt{40}}.

5

Consider the lines L_{1}: x + 2 y + 5 = 0 and L_{2}: x + b y + 20 = 0.

a

Find the distance between L_{1} and the origin.

b

Solve for the value(s) of b such that L_{1} is twice as far from the origin as L_{2}.

Geometric applications
6

Consider the circle represented by the equation x^{2} + y^{2} = 0.81.

a

Find the centre of the circle.

b

Find the radius of the circle.

c

Find the perpendicular distance from the origin to the line - 4 x + \left( - 7 y\right) + 8 = 0.

d

State the number of intersections between the line and circle. Explain your answer.

7

Consider the circle represented by the equation \left(x - \left( - 3 \right)\right)^{2} + \left(y - \left( - 2 \right)\right)^{2} = 4.

a

Find the centre of the circle.

b

Find the radius of the circle.

c

Find the perpendicular distance from the centre of the circle to the line - 3 x + 5 y + \left( - 5 \right) = 0.

d

State the number of intersections between the line and circle. Explain your answer.

8

The three points A(- 3, - 4), B(0, - 10) and C(- 6, - 7) form a triangle.

a

Find the gradient of the line AC.

b

Find the equation of the line AC in general form.

c

Find the midpoint, D, of the line AC.

d

Find the gradient of the line BD.

e

Is BD perpendicular to AC?

f

What type of triangle is ABC?

g

Find the length of the line BD. Give your answer in simplest surd form.

h

Find the length of the line AC. Give your answer in simplest surd form.

i

Find the area of triangle ABC.

j

Find the coordinates of the point E that would make ABCE a parallelogram.

9

The points A \left(5, 7\right), B \left(9, 10\right), C \left(6, 4\right) and D \left(2, 1\right) are the vertices of a parallelogram.

a

Determine the equation of the line going through AB.

b

If AB is the length of the parallelogram, find the perpendicular height of the parallelogram.

c

Determine the exact area of the parallelogram.

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Outcomes

0580E6.2D

Know that the perpendicular distance from a point to a line is the shortest distance to the line.

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