If the normal at any point `P` on the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` meets the axes at `Ga n dg,` respectively, then find the raio `P G: Pgdot`

Options

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

Answer

C

Explanation

Let P `( a cos theta, b sin theta ) ` be a point on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
the equation of the normal at P is
` ax sec theta - by " cosec" theta =a^(2)-b^(2)`.
It meets axes at
`G((a^(2)-b^(2))/(a) cos theta,0) and g(0, -(a^(2)-b^(2))/(a) sin theta)`
`therefore PG^(2)=(a cos theta -(a^(2)-b^(2))/(a) cos theta)^(2)+b^(2) sin ^(2) theta`
`implies PG^(2)=(b^(2))/(a^(2))(b^(2)cos^(2) theta +a^(2)sin^(2) theta)`
` and Pg^(2) (a^(2))/(b^(2))(b^(2) cos^(2) theta +a^(2)sin ^(2)theta)`
`implies PG"Pg =b^(2):a^(2)`