# How do you write an equation of a parabola with directrix x+y=1 and focus (1,1)?

Sep 22, 2016

${x}^{2} + {y}^{2} - 2 x y - 2 x - 2 y + 3 = 0$ graph{x^2+y^2-2xy-2x-2y+3=0 [-10, 10, -5, 5]}

#### Explanation:

We use the Focus-Directrix Property of Parabola (FDP) :

(FDP) : Let pt. $S \text{, and, line } d$ be the Focus and Directrix of a

Parabola, resp. If $P$ is any pt. on the parabola, then, the pt.$P$ is

equidistant from the pt. $S$ and the line $d$.

Recall that the $\bot \text{-dist. of a pt."P(x_0,y_0)}$ from a line :

$a x + b y + c + 0$ is given by $| a {x}_{0} + b {y}_{0} + c \frac{|}{\sqrt{{a}^{2} + {b}^{2}}} .$

Let $P \left(x , y\right)$ be any pt. on the Parabola. Then, by the discussion

above, we have,

$\sqrt{{\left(x - 1\right)}^{2} + {\left(y - 1\right)}^{2}} = | x + y - 1 \frac{|}{\sqrt{{1}^{2} + {1}^{2}}}$. Squaring both sides,

$\therefore 2 \left({x}^{2} + {y}^{2} - 2 x - 2 y + 2\right) = {x}^{2} + {y}^{2} + {\left(- 1\right)}^{2} + 2 x y - 2 y - 2 x$, or,

${x}^{2} + {y}^{2} - 2 x y - 2 x - 2 y + 3 = 0$