Find the area of the locus of center of circle formed given the following conditions

Question

$A$ is a fixed point on a circle of radius $1$ unit, $P$ is a variable point on this circle. A circle is formed touching the $CP,CA$ and the $Arc(AP)$ where $C$ is the center of the original circle. What will be the area enclosed by the locus of center of new circle so formed?

My Approach: Shifting the origin and rotating the axis. I consider the origin of the original circle to be at the origin and point A to be fixed at (1,0). Now the point P can be written as $(\cos{\theta} , \sin{\theta})$ The center of circle touching the radii CP and CA should lie on the angle bisector. Hence if $(h,k)$ is the center of this circle and $r$ is its radius then $|k|=r$ and $\sin{\theta} = \frac{r}{1-r}$ and $h = (1-r)\cos{\theta}$ using this I obtain the locus of the curve but I am not sure how to proceed further.

Also there could probably be a much better way to approach this. All hints/explanations/solutions are welcome. Thanks!

Answer 1

Let $C=(0,0)$, $A=(0,1)$.

Due to the symmetry it's enough to consider just the first quadrant, for which $x$-coordinate of the inscribed circle is $r$, and the $y$-coordinate is $y(r)=\sqrt{1-2r}$, so the area is \begin{align} S&=4\int_0^{1/2}\sqrt{1-2r}\, dr =\frac43 . \end{align}