# What is the standard form of the equation of the parabola with a directrix at x=6 and a focus at (9,5)?

Jun 28, 2017

$6 x = {y}^{2} - 10 y + 70$

#### 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 same.

Let the point on parabola be $\left(x , y\right)$. Here focus is $\left(9 , 5\right)$ and its distance from focus is $\sqrt{{\left(x - 9\right)}^{2} + {\left(y - 5\right)}^{2}}$.

And as directrix is $x = 6$ and distance of $\left(x , y\right)$ from $x = 6$ is $| x - 6 |$. Hence equation of parabola is

${\left(x - 9\right)}^{2} + {\left(y - 5\right)}^{2} = {\left(x - 6\right)}^{2}$

or ${x}^{2} - 18 x + 81 + {y}^{2} - 10 y + 25 = {x}^{2} - 12 x + 36$

or ${\cancel{x}}^{2} + 81 + {y}^{2} - 10 y + 25 - 36 = {\cancel{x}}^{2} + 18 x - 12 x$

or $6 x = {y}^{2} - 10 y + 70$

or $x = \frac{1}{6} \left({y}^{2} - 10 y + 70\right)$

graph{(y^2-10y+70-6x)(x-6)((x-9)^2+(y-5)^2-0.03)=0 [-0.83, 19.17, -0.36, 9.64]}
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