If `C` is the center and `A ,B` are two points on the conic `4x^2+9y^2-8x-36 y+4=0` such that `/_A C B=pi/2,` then prove that `1/(C A^2)+1/(C B^2)=(13)/(36)dot`

Options

A. `13/36`
B. `36/13`
C. `16/33`
D. `33/16`

Answer

A

Explanation

We have,
`4x^(2) + 9y^(2) - 8x - 36y + 4 = 0`
`rArr 4(x - 1)^(2) + 9(y - 2)^(2) = 36`
`rArr ((x-1)^(2))/(3^(2)) + ((y - 2)^(2))/(2^(2)) = 1" "` …(i)
Clearly, the coordinates of C are (1,2).
Suppose CA makes an angle `theta` with the major axis.
Then, the coordinates of A and B are
`A -= (1 + CA cos theta, 2 + CA sin theta)`
`B -= (1 + CB cos (pi/2 + theta), 2 + CB sin (pi/2 + theta))`
`-= (1 - CB sin theta, 2 + CB cos theta)`
Since A and B lie on (i). Therfore,
`(CA^(2))/(9) cos^(2)theta + (CA^(2))/(4) sin^(2)theta = 1`
and,
`(CB^(2))/(9) sin^(2)theta + (CB^(2))/(4) cos^(2)theta = 1`
`rArr 1/36(4 cos^(2)theta + 9 sin^(2)theta) = CA^(-2)`
and,
`1/36(4 sin^(2)theta + 9 cos^(2)theta) = CB^(-2)`
`rArr CA^(-2) + CB^(-2) = 13/36`