Question

Obtain the equation of a parabola with focus (4,-6) and directrix as y=-2

Answer 1

A parabola is defined as follows:

For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

Let $P(x,y)$ be the point on the parabola. Then

$$(x-4)^2+(y+6)^2=(y-(-2)^2$$

$$(x-4)^2=-y^2-12y-36+y^2+4y+4$$

$$(x-4)^2=-8y-32$$

Therefore the required equation of the parabola is

$$(x-4)^2=-8y-32$$