# What is the standard form of the equation of the parabola with a directrix at x=23 and a focus at (5,5)?

The equation of parabola will be: ${\left(y - 5\right)}^{2} = - 36 \left(x - 14\right)$
Given equation of directrix of parabola is $x = 23$ & the focus at $\left(5 , 5\right)$. It is clear that it is a horizontal parabola with sides diverging in -ve x-direction. Let general equation of parabola be
${\left(y - {y}_{1}\right)}^{2} = - 4 a \left(x - {x}_{1}\right)$ having equation of directrix: $x = {x}_{1} + a$ & the focus at $\left({x}_{1} - a , {y}_{1}\right)$
Now, comparing with given data, we have ${x}_{1} + a = 23$, ${x}_{1} - a = 5 , {y}_{1} = 5$ which gives us ${x}_{1} = 14 , a = 9$ hence the equation of parabola will
${\left(y - 5\right)}^{2} = - 4 \setminus \cdot 9 \left(x - 14\right)$
${\left(y - 5\right)}^{2} = - 36 \left(x - 14\right)$