If the conics whose equations are `S_1:(sin^2theta)x^2+(2htantheta)x y+(cos^2theta)y^2+32 x+16 y+19=0` `S_2:(cos^2theta)x^2-(2h^(prime)cottheta)x y+(sin^2theta)y^2+16 x+32 y+19=0` intersect at four concyclic points, where `theta[0,pi/2],` then the correct statement(s) can be (a)`h+h^(prime)=0` (b) `h-h^(prime)=0` (c)`theta=pi/4` (d) none of these

Options

A. `h+h'=0`
B. `h= h'`
C. `h+h'=1`
D. none of these

Answer

1

Explanation

The curve passing through the point of intersection of S and S is
`S+ lambda S' =0`
or `x^(2)(sin^(2)theta+lambdacos^(2)theta)+y^(2)(cos^(2) theta+lambdasin^(2)theta)+2xy(h+lambdah')+x(32+16lambda)+y(16+32lambda)+19(1+lambda)=0`
For this equation to be a circle.
`sin^(2)theta+lambda cos^(2)theta=cos^(2)theta+lambdasin^(2)theta` or `lambda=1`
and `h+lambdah' =0` or `h +h'=0`