# How do you find the equation in standard form of the parabola Focus (3,4) and a directrix y = 1?

Jun 13, 2016

${x}^{2} - 6 x - 6 y + 24 = 0$

#### Explanation:

We use Focus-Directrix Property of Parabola :
If Focus is $S \left(h , k\right)$ & eqn. of Directrix is $a x + b y + c = 0$, eqn. of parabola is ${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {\left[\frac{a x + b y + c}{\sqrt{{a}^{2} + {b}^{2}}}\right]}^{2} = {\left(a x + b y + c\right)}^{2} / \left({a}^{2} + {b}^{2}\right)$

In our case, it is,
${\left(x - 3\right)}^{2} + {\left(y - 4\right)}^{2} = {\left(0 \cdot x + 1 y - 1\right)}^{2} / \left({0}^{2} + {1}^{2}\right) = {\left(y - 1\right)}^{2}$
$\therefore {x}^{2} - 6 x + 9 - 8 y + 16 + 2 y - 1 = 0$
$\therefore {x}^{2} - 6 x - 6 y + 24 = 0$