Question

A line through origin meets the line x = 3y + 2 at right angles at point X. Find the co-ordinates of X.

Answer 1

The given line is

x = 3y + 2   ...(1)

3y = x – 2

`y = 1/3 x - 2/3`

Slope of this line is `1/3`

The required line intersects the given line at right angle.

∴ Slope of the required line = `(-1)/(1/3) = -3`

The required line passes through (0, 0) = (x₁, y₁)

The equation of the required line is

y – y₁ = m(x – x₁)

y – 0 = –3(x – 0)

3x + y = 0    ...(2)

Point X is the intersection of the lines (1) and (2).

Using (1) in (2), we get,

9y + 6 + y = 0

`y = (-6)/10 = (-3)/5`

∴ x = 3y + 2

= `(-9)/5 + 2`

= `1/5`

Thus, the co-ordinates of the point X are `(1/5, (-3)/5)`