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

$$A$$ 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!

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}
• How does the y coordinate comes out to be $y(r)=\sqrt{1-2r}$ ? Jun 10, 2021 at 14:21
• @protus: $|CO|=1-r$ is hypotenuse, $r$ is one leg, what is another? Jun 10, 2021 at 14:24