Find the center & radius of the largest circle touching two parabolas $y^2=4x$ and $x^2=4y$ in the bounded region.
Since the two parabolas are reflections of each other about the y=x line, the center of the circle must lie on the y=x line. Assuming it (h,h) and considering tangency is very lengthy. Is there a shorter and better approach?
Solved it actually. Assuming a point $(t^2,2t)$ and taking maximum distance of y=x gives t=1, point of contact (1,2), center (3/2,3/2) and radius $\frac{1}{\sqrt2}$.
If the tangent circle with center $O(h,h)$ touches the parabola $y=2\sqrt{x}$ at $P_1(x_1,x_2)$ then it touches the parabola $y=\frac{x^2}{4}$ at $P_2(x_2,x_1)$. Right?
For the largest circle, the slope of the tangent line at $P_1(x_1,x_2)$ which is $\frac{1}{\sqrt{x_1}}$ must be equal to $1$. Right? So, we have $P_1=(1,2)$, $P_2=(2,1)$ and their average $O=(\frac{3}{2},\frac{3}{2})$ ($h=\frac{3}{2}$).
The circle is $(x-\frac{3}{2})^2+(y-\frac{3}{2})^2=\frac{1}{2}.$