Consider an ellipse with major and minor axes of length $10$ and $8$ respectively. The radius of the largest circle that can be inscribed in this ellipse, given that the centre of this circle is one of the focus of the ellipse.
I attempted to solve this by using a very simple concept:
All points of the circle should either lie inside or on the circle. Hence assuming any point on the circle to be $x=ae+r\cos \theta$ and $y=r\sin \theta$ which satisfies:
I got a quadratic in $\cos \theta$ and I made the equation to be true independent of theta which gave $r\in [a-ae,a+ae]$ what is wrong to solve it by this method.
I would also like to know any other methods to solve this problem.