Prove that if any tangent to the ellipse is cut by the tangents at the endpoints of the major axis at `Ta n dT '` , then the circle whose diameter is `T T '` will pass through the foci of the ellipse.

Explanation

Let the ellipse be
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1,agtb
(x)/(a)cos theta+(y)/(b)sin theta=1" "(1)`
Tramgemt at the vartices are x=a and x=-a
Sloving with (1) we, get the points T and T' respectively ,as
`T(a,(b(1-cos theta))/(sin theta)),T'(-a(b(1+cos theta))/(sin theta))`
`i.e.,T(a,b tan.(theta)/(2)),T'(-a b cot.(theta)/(2))`
Circle on T'T as diameter is
`(x-a)(x+a)+(y-b tan.(theta)/(2))(y-b cot. (theta)/(2))0`
`or x^(2)-a^(2)+y^(2)+b^(2)-b[ tan.(theta)/(2)+cot.(theta)/(2)]y=0`
It will pass through the foci `(+-ae,0)`
If `a^(2)e^(2)-a^(2)+b^(2)=0`, then
`b^(2)=a^(2)(1-e^(2))`
Which is true