# What is the standard form of the equation of the parabola with a directrix at x=5 and a focus at (11,-7)?

May 22, 2016

${\left(y + 7\right)}^{2} = 12 \cdot \left(x - 8\right)$

#### Explanation:

Your equation is of the form

${\left(y - k\right)}^{2} = 4 \cdot p \cdot \left(x - h\right)$

The focus is $\left(h + p , k\right)$

The directrix is $\left(h - p\right)$

Given the focus at $\left(11 , - 7\right) \to h + p = 11 \text{ and } k = - 7$

The directrix $x = 5 \to h - p = 5$

$h + p = 11 \text{ " (eq. 1)}$
$h - p = 5 \text{ } \left(e q . 2\right)$

$\underline{\text{use (eq. 2) and solve for h}}$

$\text{ " h=5+p" (eq. 3)}$

$\underline{\text{Use (eq. 1) + (eq. 3) to find the value of } p}$

$\left(5 + p\right) + p = 11$

$5 + 2 p = 11$

$2 p = 6$

$p = 3$

$\underline{\text{Use (eq.3) to find the value of } h}$

$h = 5 + p$

$h = 5 + 3$

$h = 8$

$\text{Plugging the values of " h, p " and " k " in the equation "(y-k)^2=4*p*(x-h) " gives}$

${\left(y - \left(- 7\right)\right)}^{2} = 4 \cdot 3 \cdot \left(x - 8\right)$

${\left(y + 7\right)}^{2} = 12 \cdot \left(x - 8\right)$