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

1 Answer

#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#