Question

A straight line is passing through the point A(1, 2) with slope `5/12`. Find points on the line which are 13 units away from A

Answer 1

Slope of the line m = tan θ = `5/12`

sin θ = `5/13`

cos θ = `12/13`

The parametric equation of the line passing through the point (1, 2) making angle θ with x-axis is

`(x - 1)/cos theta` = r

`(y - 2)/sin theta` = r

Any point on this line is

`(x - 1)/costheta` = r, `y - 2)/sintheta` = r

x – 1 = r cos θ, y – 2 = r sin θ

x = 1 + r cos θ, y = 2 + r sin θ

(1 + r cos θ, 2 + r sin θ)    ......(1)

Where r is the distance of any point from A(1, 2) on the line.

To find the point which is 13 units away from A(1, 2) on the line.

Substitute r = ± 13, cos θ =  `12/13`, 

sin θ = `5/13` in equation (1)

Required point = `(1 +- 13 (12/13), 2 +- 13 (5/13))`

= (1 ± 12, 2 ± 5)

= (1 + 12, 2 + 5)

= (1 + 12, 2 + 5) or (1 – 12, 2 – 5)

= (13, 7) or (– 11, – 3)