From point `P(8,27)` tangents PQ and PR are drawn to the ellipse `(x^(2))/(4)+(y^(2))/(9)=1.` Then, angle subtended by QR at origin is

Options

A. `tan ^(-1)""(2sqrt6)/(65)`
B. `tan ^(-1)""(4sqrt6)/(65)`
C. `tan^(-1)""(8sqrt2)/(65)`
D. None of these

Answer

D

Explanation

Equation of chord of contact QR is
`8*x/4+27*y/9=1`
`implies2x+3y=1" "...(i)`
Now, equation of the pari of lines passing through origin and points Q, R given by
`((x^(2))/(4)+(y^(2))/(9))=(2x+3y)^(2)`
`implies9x^(2)+4y^(2)=36(4x^(2)+12xy+9y^(2))`
`implies135x^(2)+432xy+320y^(2)=0`
`therefore` Required angle is
`=tan ^(-1)""(2sqrt((216)^(2)-135*320))/(455)`
`=tan ^(-1)""(8sqrt(2916-2700))/(455)`
`=tan ^(-1)""(8sqrt216)/(455)`
`=tan ^(-1)""(48sqrt6)/(455)`