If P and P denote the length of the perpendicular from a focus and the centre of an ellipse with semi - major axis of length a, respectively , on a tangent to the ellipse and r denotes the focal distance of the point , then

Options

A. `ap= rp'`
B. `rp= ap'`
C. `ap=rp'+1`
D. `ap' +rp=1`

Answer

A

Explanation

Let the ellipse be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` and let
`( x cos theta )/( a) +(y sin theta )/(b)=1`
be a tangent to it at point (a cos ` theta , b sin theta )` then
P = Length of the perpendicular from s(ae,0) on (i)
`implies P=|(e-cos theta-1)/(sqrt((cos^(2) theta )/(a^(2)))+( sin ^(2) theta )/(b^(2)))|`
P' = length of the perpendicular from O(0,0) on (i)
`implies p'=|(1)/(sqrt((cos^(2) theta)/(a^(2))+( sin ^(2) theta)/(b^(2))))|`
and ,r=ae cos `theta -a `
Clearly ,rp'=ap