Question

What is the equation of the parabola with a focus at (15,-3) and a directrix of y= -4?

Answer 1

Equation of parabola is `x^2-30x-2y+218=0`

Explanation:

Here the directrix is a horizontal line `y=-4`.

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 `(15,-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-15)^2+(y+3)^2)` and from directrix will be `|y+4|`

Hence, `(x-15)^2+(y+3)^2=(y+4)^2`

or `x^2-30x+225+y^2+6y+9=y^2+8y+16`

or `x^2-30x-2y+234-16=0`

or `x^2-30x-2y+218=0`