Dynamic Geometry: P120

From the compiled calculations in Dynamic Geometry: P96 Series , we know that the radius of the circle is $6.5$ and the red dividing chord $AB=12$ . If the center $O$ of the circle is the origin $(0,0)$ of the $xy$ -plane, where $AB$ is parallel to the $x$ -axis, then $AB$ is along $y=2.5$ .

Let an arbitrary point on the locus or the center of the moving semicircle be $P(x,y)$ . Note the upper and lower parts of the locus is continuous and it is traced by $P$ . Note the $OP$ is along the diameter $CC'$ . Let $MN$ be the diameter of the semicircle and $CP=k$ . By intersecting chords theorem ,

$\begin{aligned} MP \cdot PN & = CP \cdot PC' \\ r^2 & = k(13-k) \end{aligned}$

Let $PQ$ be perpendicular to the $x$ -axis. By Pythagorean theorem ,

$\begin{aligned} OQ^2 + PO^2 & = OP^2 \\ x^2 + y^2 & = (6.5 - k)^2 \\ x^2 + y^2 & = \frac {169}4 \blue{- 13k + k^2} & \small \blue{\text{Note that }r^2 = 13k - k^2} \\ x^2 + y^2 & = \frac {169}4 \blue{- r^2} & \small \blue{\text{and }y = PQ = r + 2.5} \\ x^2 + y^2 & = \frac {169}4 - \blue{\left(y-\frac 52\right)^2} \\ x^2 + y^2 & = \frac {169}4 - y^2 + 5y - \frac {25}4 \\ x^2 + 2y^2 - 5y & = 36 \\ x^2 + 2 \left(y - \frac 54\right)^2 & = 36 + \frac {25}8 = \frac {313}8 \\ \frac {x^2}{\frac {313}8} + \frac {\left(y-\frac 54\right)^2}{\frac {313}{16}} & = 1 \end{aligned}$

Therefore the locus is an ellipse with center at $\left(0, \dfrac 54\right)$ , a major semi-axis $a = \sqrt{\dfrac {313}8}$ , a minor semi-axis $b = \dfrac {\sqrt{313}}4$ , and area $ab\pi = \dfrac {313\pi}{8\sqrt 2} \approx \boxed {87}$ .