If the line Ix + my +n=0 cuts the ellipse `(x^(2))/(a^(2))+(Y^(2))/(b^(2))=1` in point eccentric angles differ by `pi//2`, then

Options

A. `a^(2)l^(2)+b^(2)m^(2)=2n^(2)`
B. `a^(2)l^(2)+b^(2)m^(2)=n^(2)`
C. `a^(2)m^(2)+b^(2)l^(2)=2n^(2)`
D. `a^(2)m^(2)+b^(2)l^(2)=n^(2)`

Answer

A

Explanation

suppose the line lx+my +n=0 cuts the ellipse at P( a cos ` theta, b sin theta ) and Q ( a cos (pi//2+theta),b sin (pi//2+theta)),`
then these two point lie on the given line .
`therefore la cos theta + m b sin theta + n =0`
`and -la sin theta + m b sin cos theta + n=0`
`implies la cos theta + mb sin theta =- n and ,-la sin theta + mb cos theta =-n`
`implies (la cos theta + mb sin theta)^(2) +(-la sin theta +mb cos theta )^(2)=n^(2)+n^(2)`
`implies l^(2) a^(2) +m^(2)b^(2)=2n^(2)`