locus of intersection of lines, variable diameter related

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#1 h Aug 20, 2021, 11:09 PM • 1 Y

Y by jhu08

Let $\omega$ be a circle and $A$ and $B$ be two points on $\omega$ such that segment $AB$ is not a diameter. If $XY$ is a variable diameter of the same circle, determine the locus of the intersection point of the lines $AX$ and $BY$ .

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#2 h Aug 21, 2021, 2:55 PM

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Given the circle, midpoint $O(0,0)$ and radius $r$ .

Points $X(r\cos \alpha,r\sin \alpha)$ and $Y(-r\cos \alpha,r\sin \alpha)$ with $\alpha$ fixed.
Points $A(r\cos \lambda,r\sin \lambda)$ and $B(-r\cos \lambda,-r\sin \lambda)$ with $\lambda$ variable.

The lines $AX$ and $BY$ cut in the point $P(-r \cot \alpha \sin \lambda,\frac{r(1+\cos \alpha\cos \lambda)}{\sin \alpha}\ )$ .
The points $P$ lie on the circle $x^{2}+y^{2}-2\frac{r}{\sin \alpha} \cdot y +r^{2}=0$ .

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