Consider an ellipse whose major and minor axis of length $10$ and $8$ unit respectively. Then the radius of the largest circle which can be inscribed in such an ellipse if the circle's center is one focus of the ellipse.
What I tried:
Assuming that major axis and minor axis of an ellipse along coordinate axis.($x$ and $y$ axis respectively)
Then equation of ellipse is $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ where $a=5,b=4$
Then $\displaystyle b^2=a^2(1-e^2)$ . getting $\displaystyle e=3/5.$
coordinate of focus is $(\pm ae,0)=(\pm 3,0)$
Let equation of circle is $(x\pm 3)^2+y^2=r^2$
How do I solve it from here?
Best Answer
Denote $F, F'$ the foci of the ellipse. Assume we want to find the greatest circle inscribed in the ellipse, with center $F.$
The circle and the ellipse have a common tangent line and also a common normal in their point (points) of tangency $P,$ eventually also $P'$ if they are two.
The normal of an ellipse bisects the angle between the lines to the foci. In our case, the normal at $P$ bisects the angle between $PF$ and $PF'.$
The normal of a circle passes through its center.
From the above properties follows that $P,F,F'$ are collinear. Therefore, $P$ is the vertex of the ellipse lying at the major axis nearest to $F,$ and the equation of the circle is $$(x-3)^2+y^2=|PF|^2=4$$