Circle and Locus _ ONLY PEN AND PAPER ALLOWED. Q) Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let $F_{1}$ be the set of all pairs of circles $(S_{1}$, $S_{2}$) such that T is tangent to $S_{1}$ at P and tangent to $S_{2}$ at Q, and also such that $S_{1}$ and $S_{2}$ touch each other at a unique point, say, M. Let $E_{1}$ be the set representing the locus of M as the pair ($S_{1}$, $S_{2}$) varies in $F_{1}$. Let the set of all straight line segments joining a pair of distinct points of $E_{1}$ and passing through the point R(1, 1) be $F_{2}$. Let $E_{2}$ be the set of the mid-points of the line segments in the set $F_{2}$. Let $C$ be the circle $x^2+y^2+6(2y+7x)=53$. The number of times $C$ intersects $E_{1}$ and $E_{2}$ is (are):
As the diameter of the first is $PQ$ $$E_1=\{(x,y)| (x+2)(x-2)+(y-7)(y+5)=0\}-\{P,Q\}$$

and the diameter of the second is $DR$, with $D$ the midpoint of $PQ$
$$E_2=\{(x,y)| x(x-1)+(y-1)^2=0\}-\{(4/5,7/5),(36/37,43/37)\}$$

where the disallowed points stem from the intersections with $l_{PR}$ and $l_{QR}$.

Now

$$\#((V((x+2)(x-2)+(y-7)(y+5))\cup V(x(x-1)+(y-1)^2))\cap C)=4$$

but three points are disallowed, leaving the other intersection $(400/409,349/409)$ so

$$\#((E_1\cup E_2)\cap C)=1$$



Edit

$E_2$ is part of a circle: let $y-1=m(x-1)$ be lines through $R$, then together with the equation for the circle $E_1$ is a part of, we get the relation $(m^2+1)y^2+(-2m^2+2m-2)y-38m^2-2m+1=0$ that gives us solutions $y_{1,2}$ with corresponding $x_{1,2}$. What we're looking for are points $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})=(\frac{1}{m^2+1},\frac{m^2-m+1}{m^2+1})$ which implicitizes to $x^2-x+y^2-2y+1=0.$
Let the centers of $S_1$ and $S_2$ be $O_1$ and $O_2$ respectively. Note that $O_1P\perp PQ$ and also $O_2Q\perp PQ$.

Since $O_1M=O_1P$ and $O_2M=O_2Q$ then both $O_1PM$ and $O_2QM$ are isosceles triangles and
$$\begin{align}\angle MPO_1&amp;=\angle PMO_1\\ \angle MQO_2&amp;=\angle QMO_2\end{align}$$
This implies
$$\angle PMQ=\angle MPQ+\angle QPM$$
which means that $\angle PMQ$ is a right angle. So $M$ lies on a circle with diameter $PQ$ (that is called the set $E_1$).

The center of $E_1$ is located at $D(0,1)$ and its radius is $2\sqrt{10}$:
$$E_1=\{(x,y):x^2+(y-1)^2=40\}$$

Now imagine the set of all the line segments passing through $R(1,1)$ and crossing $E_1$ at two points $A$ and $B$, and note that the point $R$ is inside $E_1$ and $AB$ is in fact a chord of the $E_1$ circle.

Can you show that the loci of the midpoints of all these chords is a circle whose diameter is $RD$?