Question

What is the equation in standard form of the parabola with a focus at (-4,-1) and a directrix of y= -3?

Answer 1

THe equation of the parabola is `(x+4)^2=4(y+2)`

Explanation:

The focus is `F=(-4,-1)`

The directrix is `y=-3`

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

Therefore,

`(y+3)^2=(x+4)^2+(y+1)^2`

`cancel(y^2)+6y+9=(x+4)^2+cancel(y^2)+2y+1`

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

`(x+4)^2=4y+8=4(y+2)`

graph{((x+4)^2-4y-8)(y+3)((x+4)^2+(y+1)^2-0.01)=0 [-10, 10, -5, 5]}