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

Question

Ethan Adkins · Accepted Answer

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

A parabola is the locus of a point that moves in such a way that its distance from the focus point and its distance from the directrix line are 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|#

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

Eva Abramson · Answer

undefined The equation of the parabola is $ (x - h)^2 = 4p(y - k) $, where (h, k) is the vertex and p is the distance from the vertex to the focus (or directrix). Given that the focus is at (9,12) and the directrix is y = -13, the vertex is (9, -0.5) and the distance from the vertex to the focus (or directrix) is 12.5. Thus, the equation of the parabola is $ (x - 9)^2 = 50(y + 0.5) $.