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

Aug 7, 2018

$x = \frac{1}{8} {\left(y + 1\right)}^{2} - 8$

#### Explanation:

Parabola is the locus of a point which moves so that its distance from a given point called focus and a given line called directrix is always equal.

Let the point be $\left(x , y\right)$. Its distance from focus $\left(1 , - 1\right)$ is

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

and its distance from directrix $x = - 3$ or $x + 3 = 0$ is $x + 3$

Hence equation of parabola is $\sqrt{{\left(x - 1\right)}^{2} + {\left(y + 1\right)}^{2}} = x + 3$

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

i.e. ${x}^{2} - 2 x + 1 + {y}^{2} + 2 y + 1 = {x}^{2} + 6 x + 9$

i.e. ${y}^{2} + 2 y - 7 = 8 x$

or $8 x = {\left(y + 1\right)}^{2} - 8$

or $x = \frac{1}{8} {\left(y + 1\right)}^{2} - 8$

graph{(y^2+2y-7-8x)((x-1)^2+(y+1)^2-0.01)(x+3)=0 [-11.17, 8.83, -5.64, 4.36]}