- 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
- Find a Cartesian equation for the conic section consisting of all points $(x, y)$ whose distance from the point $(0, 2)$ is half the distance from the line $y=8$
- Given a line $L,$ a point $F$ not on $L,$ and a positive number $e.$ Let $d(X,L)$ denote the distance from a point $X$ to $L.$ The set of all $X$ satisfying the relation $$\|X - F\| = ed(X,L)$$is called a conic section with eccentricity $e.$ If $N$ is a vector normal to $L$ and if $P$ is any point on $L,$ the distance $d(X, L)$ from any point $X$ to $L$ is given by the formula $$d(X, L) = \frac{\left|(X - P) \cdot N\right|}{\|N\|}$$
- Let $F = (0, 2),$ $P = (0, 8),$ and $N = (0, 1).$ The conic section consisting of all points $X$ whose distance from $F$ is half the distance from $P$ is the set of $X$ satisfying the relation $$\|X - F\| = \frac{1}{2}\left|(X - P) \cdot N\right|$$Putting the vectors into $(x, y)$ coordinates $$\|(x, y-2)\| = \frac{1}{2}\left|(x, y - 8) \cdot (0, 1)\right|$$Squaring both sides $$x^{2} + (y - 2)^{2} = \frac{1}{4}(y - 8)^{2}$$Expanding the expressions and rearranging terms into standard form we get $$\frac{x^{2}}{12} + \frac{y^{2}}{16} = 1\quad \blacksquare$$