Question

What is the equation of the parabola with a focus at (-1,3) and a directrix of y= -6?

Answer 1

Equation of parabola is `x^2+2x-18y-26=0`

Explanation:

Here the directrix is a horizontal line `y=-6`.

Since this line is perpendicular to the axis of symmetry, this is a regular parabola, where the `x` part is squared.

Now the distance of a point on parabola from focus at `(-1,3)` is always equal to its between the vertex and the directrix should always be equal. Let this point be `(x,y)`.

Its distance from focus is `sqrt((x+1)^2+(y-3)^2)` and from directrix will be `|y+6|`

Hence, `(x+1)^2+(y-3)^2=(y+6)^2`

or `x^2+2x+1+y^2-6y+9=y^2+12y+36`

or `x^2+2x-18y+10-36=0`

or `x^2+2x-18y-26=0`