KöMaL 2017 A699

A circle $\omega$ lies in a circle $\Omega$ such that their common center is the point $O$. Fix a point $A\ne O$ inside $\omega$. Let $X$ denote an arbitrary point on the circumference of $\Omega$, and let $Y$ denote the second intersection point of $\Omega$ and the line $AX$. Let $Z$ denote the intersection of $\omega$ and the line segment $AX$. Let $M$ denote the point on the line segment $AZ$ for which $MX\cdot MZ\cdot AY = MA\cdot MY\cdot XZ$. Let $x$ and $y$ denote the tangents of the circle $\Omega$ at the points $X$ and $Y$, respectively. Let $t$ denote the line that passes through $M$ and either also passes through the intersection of $x$ and $y$, or is parallel to both $x$ and $y$. Finally, let $T$ denote the intersection of $t$ and the line $OZ$. Show that the locus of the points $T$, as $X$ is varied, is an ellipse, and the line $t$ is a tangent of this ellipse.