The points P,Q,R are taken on ellipse `x^2/a^2+y^2/b^2=1` with eccentric angles `theta, theta+a, theta+2a`, then area of ,`DeltaPQR` is independent of

Options

A. `a:b`
B. `b:a`
C. `a^(2):b^(2)`
D. `b^(2):a^(2)`

Answer

B

Explanation

Let P ( acos `theta _(1) ,,b sin theta_(1)) Q(a cos theta_(2) , b sin theta_(2)) And R( a cos theta_(3), bsin theta_(3))` be the vertices of the triangle inscribed in the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` the points on the auxiliary circle corresponding to these points are ` P' ( a cos theta_(1) , a sin theta_(1) ) Q ( a cos theta_(2) , a theta _(2) ) and ( a cos theta_(2) ) and R'( cos theta _(3) , a sin theta_(3)).`
` therefore Delta_(1) ="Area of "Delta PQR `
`implies Delta_(1) =(1)/(2) |{:(a cos theta_(1), asin theta_(1),1),(a cos theta_(2), a sin theta_(2),1),(a cos theta_(3) , b sin theta_(3),1):}|=(1)/(2) ab |{:(cos theta_(1), sin theta_(1) ,1),( cos theta_(2), sin theta_(2),1),( cos theta_(3) sin theta_(3),1):}|`
` and , `
`Delta_(2) ="Area of "Delta P'Q'R'`
`implies Delta_(1) =(1)/(2) |{:(a cos theta_(1), asin theta_(1),1),(a cos theta_(2), a sin theta_(2),1),(a cos theta_(3) , b sin theta_(3),1):}|=(1)/(2) a^(2) |{:(cos theta_(1), sin theta_(1) ,1),( cos theta_(2), sin theta_(2),1),( cos theta_(3) sin theta_(3),1):}|`
Clearly , `( Delta _(1))/(Delta_(2))= (b)/(a)=` constant .