# Obtain the equation of the parabola with focus (3,2) and directrix 3x-4y+9=0?

Apr 20, 2018

$16 {x}^{2} + 9 {y}^{2} + 24 x y - 204 x - 28 y + 244 = 0$

#### Explanation:

You suppose a point on the parabola $\left(x , y\right)$

The distance between the point $\left(x , y\right)$ and the directrix is the same distance from the point $\left(x , y\right)$ to the focus

$\sqrt{{\left(x - 3\right)}^{2} + {\left(y - 2\right)}^{2}} = | \setminus 3 x - 4 y + 9 \frac{|}{\sqrt{{3}^{2} + {\left(- 4\right)}^{2}}}$

now by squaring both sides

${\left(x - 3\right)}^{2} + {\left(y - 2\right)}^{2} = {\left(3 x - 4 y + 9\right)}^{2} / 25$

Simplify

$25 \left({x}^{2} - 6 x + 9 + {y}^{2} - 4 y + 4\right) = {\left(3 x + 9\right)}^{2} + \left(2\right) \left(- 4 y\right) \left(3 x + 9\right) + 16 {y}^{2}$

now multiply and simplify you get the following

$16 {x}^{2} + 9 {y}^{2} + 24 x y - 204 x - 28 y + 244 = 0$