Question

In the given figure, line AB meets y-axis at point A. Line through C(2, 10) and D intersects line AB at right angle at point P. Find:

1. equation of line AB.
2. equation of line CD.
3. co-ordinates of points E and D.

Answer 1

In the given figure, AB meets y-axis at point A.

Line through C(2, 10) and D intersects line AB at P at right angle.

i. Slope of AB (m) = `(y_2 - y_1)/(x_2 - x_1)`

= `(8 - 6)/(-6 - 0)`

= `2/(-6)`

= `(-1)/3`

∴ Equation of line AB is given by

y – y₁ = m(x – x₁)

`\implies y - 6 = (-1)/3(x - 0)`

`\implies` 3y – 18 = –x

`\implies` x + 3y – 18 = 0

`\implies` x + 3y = 18   ...(1)

ii. ∵ CD ⊥ AB

∴ Slope of line CD = `-(3/(-1))` = 3  ...[∵ m₁m₂ = –1]

∴ Equation of CD is given by

y – 10 = 3(x – 2)

`\implies` y – 10 = 3x – 6

`\implies` 3x – y + 10 – 6 = 0

`\implies` 3x – y + 4 = 0   ...(2)

iii. Since equation (2) meets x-axis at D

∴ Putting y = 0 in 3x – y + 4 = 0

`\implies` 3x – 0 + 4 = 0

`\implies` 3x + 4 = 0

`\implies` 3x = −4

`\implies x = (-4)/3` 

∴ Co-ordinates of D are `((-4)/3, 0)`

∵ Since equation (1) meets x-axis at E, so putting

y = 0 in x + 3y = 18

∴ x + 0 = 18 `\implies` x = 18

∴ Co-ordinates of E are (18, 0).