Question

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

Answer 1

`x^2+16y-32=0`

Explanation:

Given that the focus of parabola is at `(0, 0)` & directrix is `y=4`

The above parabola vertical downward which has vertex at `(\frac{0+0}{2}, \frac{0+4}{2})\equiv(0, 2)\equiv(x_1, y_1)` & axis of summetry as x-axis hence its equation is

`(x-x_1)^2=-4a(y-y_1)`

`(x-0)^2=-4a(y-2)`

`x^2=-4a(y-2)`

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(4)(y-2)`

`x^2=-16y+32`

`x^2+16y-32=0`