Question

What is the equation of the parabola with a focus at (3,18) and a directrix of y= -21?

Answer 1

`78y=x^2-6x-108`

Explanation:

Parabola is the locus of a pint, which moves so that its distance from a point called focus and a line called directrix is always equal.

Let the point on parabola be `(x,y)`,

its distance from focus `(3,18)` is

`sqrt((x-3)^2+(y-18)^2)`

and distance from directrix `y-21` is `|y+21|`

Hence equation of parabola is, `(x-3)^2+(y-18)^2=(y+21)^2`

or `x^2-6x+9+y^2-36y+324=y^2+42y+441`

or `78y=x^2-6x-108`

graph{(x^2-6x-78y-108)((x-3)^2+(y-18)^2-2)(x-3)(y+21)=0 [-157.3, 162.7, -49.3, 110.7]}