Definition:Joachimsthal's Equation/Circle

Definition
Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.

Let $P = \tuple {x_1, y_1}$ be an arbitrary point in the Cartesian plane.

Let $\LL$ be a straight line through $P$ which intersects $\CC$ at points $U$ and $V$.

Let $Q = \tuple {x, y}$ be a point on $\LL$.

Let $V$ divide $PQ$ in the ratio $k : 1$.

Joachimsthal's equation is the quadratic equation describing the coordinates of $U$ and $V$:


 * $k^2 \paren {x^2 + y^2 - r^2} + 2 k \paren {x x_1 + y y_1 - r^2} + \paren { {x_1}^2 + {y_1}^2 - r^2} = 0$