Question

What is the equation of the parabola with a focus at (-1,-2) and a directrix of y= -10?

Answer 1

`y=x^2/16+x/8-95/16`

Explanation:

Let `(x_0, y_0)` be a point on the parabola.
Focus of the parabola is given at `(-1, -2)`
Distance between the two points is
`sqrt((x_0-(-1))^2+(y_0-(-2))^2`
or `sqrt((x_0+1)^2+(y_0+2)^2`
Now distance between the point `(x_0,y_0)` and the given directrix `y=-10`, is
`|y_0-(-10)|`
`|y_0+10|`

Equate the two distance expressions and squaring both sides.
`(x_0+1)^2+(y_0+2)^2=(y_0+10)^2`
or `(x_0^2+2x_0+1)+(y_0^2+4y_0+4)=(y_0^2+20y_0+100)`

Rearranging and taking term containing `y_0` to one side
`x_0^2+2x_0+1+4-100=20y_0-4y_0`
`y_0=x_0^2/16+x_0/8-95/16`

For any point `(x,y)` this must be true. Therefore, the equation of the parabola is
`y=x^2/16+x/8-95/16`