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If the reflection of the ellipse ((x-4)^...

If the reflection of the ellipse `((x-4)^(2))/(16)+((y-3)^(2))/(9) =1` in the mirror line `x -y -2 = 0` is `k_(1)x^(2)+k_(2)y^(2)-160x -36y +292 = 0`, then `(k_(1)+k_(2))/(5)` is equal to

A

A. 4

B

B. 5

C

C. 6

D

D. 7

Text Solution

Verified by Experts

The correct Answer is:
B
Let `(4 cos theta+ 4,3 sin theta +3)` be any point on the ellipse `((x-4)^(2))/(16) + ((y-3)^(2))/(9) =1`
Image of `(4 cos theta + 4,3 sin theta +3)` about the line `x - y -2 =0` is (h,k), then
`(h-4 cos theta -4)/(1) = (k-3 sin theta -3)/(-1) = -(2(4 cos theta -3 sin theta-1))/(2)`
`rArr h = 3 sin theta + 5` and `k = 4 cos theta +2`

Locus of (h,k) is `((x-5)/(3))^(2) + ((y-2)/(4))^(2) =1`
`rArr 16x^(2) + 9y^(2) - 160x -36y +292 =0`
`rArr k_(1)+ k_(2) = 25`
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