Question

3.

In the figure, the straight line L:2x-y-18 = 0 cuts the x-axis and the y-axis at
A and B respectively.
L: 2x - y - 18 = 0
(a)
(i) Find the slope of L.
(ii) Find the coordinates of A and B.
(3 marks)
b)Find the equation of the perpendicular bisector of AB in general form.​

Answer 1

see explanation

Explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Rearrange 2x - y - 18 = 0 into this form

Subtract 2x - 18 from both sides

- y = - 2x + 18 ( multiply through by - 1 )

y = 2x - 18 ← in slope- intercept form

with slope m = 2 and y- intercept B(0, - 18)

To find the x- intercept let y = 0 in the equation and solve for x

2x - 18 = 0 ( add 18 to both sides )

2x = 18 ( divide both sides by 2 )

x = 9

Hence x- intercept A(9, 0)

(b)

The perpendicular bisector passes through the midpoint of AB at right angles.

Given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/(2)`

The midpoint of AB is

[0.5(9 + 0), 0.5(0 - 18) ] = (4.5, - 9)

Thus

y = -
`(1)/(2)` x + c ← is the partial equation

To find c substitute (4.5, - 9) into the partial equation

- 9 = - 2.25 + c ⇒ c = - 9 + 2.25 = - 6.75 = -
`(27)/(4)`

y = -
`(1)/(2)` x -
`(27)/(4)` ← in slope- intercept form

Multiply through by 4

4y = - 2x - 27 ( subtract - 2x - 27 from both sides )

2x + 4y + 27 = 0 ← in general form