JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Circle

Let the radius of the inscribed circle be $r$ and let $CD=\alpha$. Then $AB=2\alpha$.

The height of the trapezoid is $2r$ (diameter of the circle).

Area of trapezoid:


So $\alpha r=6$.

For the circle to be tangent to the slant side $BC$, the perpendicular distance from the center to $BC$ must equal $r$. The line $BC$ has slope $-\frac{2r}{\alpha }$ and passes through $(2\alpha ,0)$.

Using the tangency condition with the circle centered at $(r,r)$, we get $2\alpha =3r$.

From $\alpha r=6$ and $2\alpha =3r$, we get $r=2$.