# circle inside an ellipse with fixed width but variable length

Inside an ellipse with width 'a' and length 'b', there is a circle. The circle touches the point (-a,0), which means circle touches the edge of the ellipse but ellipse doesn't cut through the circle. I need to find the minimum value or the range of 'b' when the radius of the circle is 'r' and 'a'='R'. How do I do it? note: a_0=R-r

• If you were given the value of $b$ as well as $a,$ do you know how you could compute the radius of curvature of the ellipse at $(-a,0)$? – David K Jan 10 at 16:20
• Hint: Show that the center of the circle lies on a focus of the ellipse. – amd Jan 11 at 2:16

## 1 Answer

The smallest value of $$b$$ is that leading to tangency between ellipse and circle.

Write down the equations of the ellipse and circle, eliminate $$y^2$$ to get a single quadratic equation in $$x$$. Ellipse and circle are tangent when that equation has a single solution, i.e. when its discriminant vanishes.