- 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 parabola whose focus is at the origin and whose directrix is the line $x + y + 1 = 0$
- Recall that a parabola is a conic section with eccentricity equal to 1. For more information, see the hint from Exercise 7.
- The parabola whose focus is at the origin and whose directrix is the line $x + y + 1 = 0$ is the set of $X$ satisfying the relation $$\|X\| = \frac{\left|(X-P)\cdot N\right|}{\|N\|}$$where $P = (0, 1)$ and $N = (1, 1).$ Putting the vectors into $(x, y)$ coordinates and squaring both sides we get $$x^{2} + y^{2} = \frac{1}{2}(x + y+ 1)^{2}$$Expanding the right side and rearranging terms gives us $$x^{2} - 2xy + y^{2} - 2x - 2y = 1 \quad \blacksquare$$