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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.
Solution
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.
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