Question

How do you find an equation of the parabola with vertex (-8,5) and directrix y=19/4?

Answer 1

The equation is `y=2(x+8)^2+39/8`

Explanation:

Any point `(x,y)` on the parabola is equidistant from the directrix and from the focus

`(y-19/4)=sqrt((x+8)^2+(y-5)^2)`

Squaring both sides

`(y-19/4)^2=(x+8)^2+(y-5)^2`

`y^2-19/2y+361/16=(x+8)^2+y^2-10y+25`

`10y-19/2y=(x+8)^2+25-361/16`

`1/2y=(x+8)^2+39/16`

`y=2(x+8)^2+39/8`

graph{(y-2(x+8)^2-39/8)(y-19/4)=0 [-16.17, 3.83, 0.295, 10.295]}