\begin{definition}[Definition:Joachimsthal's Equation/Circle]
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 $V$:
:$k^2 \paren {x^2 + y^2 - r^2} + 2 k \paren {x x_1 + y y_1 - r^2} + \paren { {x_1}^2 + {y_1}^2 - r^2} = 0$
\end{definition}