- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
13.25. Miscellaneous review exercises on conic sections
- Show that the locus of the centers of a family of circles, all of which pass through a given point and are tangent to a given line, is a parabola.
- If you're trying to calculate it, you're doing it wrong.
- Let the given point be the focus $F,$ and the tangent line be the directrix $L.$ Then, since $F$ and the point of contact $P$ on $L$ are both on the circle, then they are both the same distance from the center of the circle. In other words, the locus of centers is the set of $X$ satisfying $$\|X - F\| = d(X, L)$$which is, by definition, a parabola. $\quad \blacksquare$