Definition:Joachimsthal's Equation
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Definition
Joachimsthal's Equation for 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 $V$:
- $k^2 \paren {x^2 + y^2 - r^2} + 2 k \paren {x x_1 + y y_1 - r^2} + \paren { {x_1}^2 + {y_1}^2 - r^2} = 0$
Also see
- Results about Joachimsthal's equation can be found here.
Source of Name
This entry was named for Ferdinand Joachimsthal.