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Question
A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. The locus of the point is ______.
Options
3x2 + 4y2 = 192
4x2 + 3y2 = 192
x2 + y2 = 192
None of these
Solution
A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. The locus of the point is 3x2 + 4y2 = 192.
Explanation:
Let (h, k) be the coordinates of the moving point.
Then, we have
`sqrt((h - 4)^2 + k^2) = 1/2 (h - 16)/sqrt(1^2 + 0)` (Why?)
(h – 4)2 + k2 = `1/4 (h - 16)^2`
4(h2 – 8h + 16 + k2) = h2 – 32h + 256
or 3h2 + 4k2 = 192
Hence, the required locus is given by 3x2 + 4y2 = 192
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