Question

What is the equation of the parabola with a focus at (9,12) and a directrix of y= -13?

Answer 1

`x^2-18x-50y+56=0`

Explanation:

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

Let the point be `(x,y)`. Its distance from focus `(9,12)` is

`sqrt((x-9)^2+(y-12)^2)`

and its distance from directrix `y=-13` i.e. `y+13=0` is `|y+13|`

hence equation is

`sqrt((x-9)^2+(y-12)^2)=|y+13|`

and squaring `(x-9)^2+(y-12)^2=(y+13)^2`

or `x^2-18x+81+y^2-24y+144=y^2+26y+169`

or `x^2-18x-50y+56=0`

graph{(x^2-18x-50y+56)((x-9)^2+(y-12)^2-1)(y+13)=0 [-76.8, 83.2, -33.44, 46.56]}