# How do you find an equation of the parabola with focus (0,0) and directrix y=4?

${x}^{2} + 16 y - 32 = 0$

#### Explanation:

Given that the focus of parabola is at $\left(0 , 0\right)$ & directrix is $y = 4$

The above parabola vertical downward which has vertex at $\left(\setminus \frac{0 + 0}{2} , \setminus \frac{0 + 4}{2}\right) \setminus \equiv \left(0 , 2\right) \setminus \equiv \left({x}_{1} , {y}_{1}\right)$ & axis of summetry as x-axis hence its equation is

${\left(x - {x}_{1}\right)}^{2} = - 4 a \left(y - {y}_{1}\right)$

${\left(x - 0\right)}^{2} = - 4 a \left(y - 2\right)$

${x}^{2} = - 4 a \left(y - 2\right)$

The directrix of above parabola is $y - {y}_{1} = a$ but given that directrix is $x = 4$ thus by comparing both the equations of directrix we get $a = 4$ hence the equation of parabola is

${x}^{2} = - 4 \left(4\right) \left(y - 2\right)$

${x}^{2} = - 16 y + 32$

${x}^{2} + 16 y - 32 = 0$