# What is the standard form of the equation of the parabola with a directrix at x=3 and a focus at (1,-1)?

May 30, 2016

${y}^{2} + 4 x + 2 y - 7 = 0$

#### Explanation:

Let their be a point $\left(x , y\right)$ on parabola. Its distance from focus at $\left(1 , - 1\right)$ is

$\sqrt{{\left(x - 1\right)}^{2} + {\left(y + 1\right)}^{2}}$

and its distance from directrix $x = 3$ will be $| x - 3 |$

Hence equation would be

$\sqrt{{\left(x - 1\right)}^{2} + {\left(y + 1\right)}^{2}} = \left(x - 3\right)$ or

${\left(x - 1\right)}^{2} + {\left(y + 1\right)}^{2} = {\left(x - 3\right)}^{2}$ or

${x}^{2} - 2 x + 1 + {y}^{2} + 2 y + 1 = {x}^{2} - 6 x + 9$ or

${y}^{2} + 4 x + 2 y - 7 = 0$

graph{y^2+4x+2y-7=0 [-11.21, 8.79, -5.96, 4.04]}