Question

Find the equation of the set of all points whose distance from (0, 4) are `2/3` of their distance from the line y = 9.

Answer 1

Let P(x, y) be a point.

We have `sqrt((x - 0)^2 + (y - 4)^2) = 2/3|(y - 9)/1|`

Squaring both sides, we have

`x^2 + (y - 4)^2 = 4/9(y^2 + 81 - 18y)`

⇒ 9x² + 9(y – 4)² = 4y² + 324 – 72y

⇒ 9x² + 9y² + 144 – 72y = 4y² + 324 – 72y

⇒ 9x² + 5y² + 144 – 324 = 0

⇒ 9x² + 5y² – 180 = 0

Hence, the required equation is 9x² + 5y² – 180 = 0.