What is the standard form of the equation of the parabola with a focus at (6,5) and a directrix of #y= -1#?

1 Answer
Aug 2, 2018

Parabola is #y=1/12(x-6)^2+2#

Explanation:

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

Let the point be #(x,y)#. Its distance from focus #(6,5)# is

#sqrt((x-6)^2+(y-5)^2)#

and its distance from directrix #y=-1# is #y+1#

Hence equation of parabola is #sqrt((x-6)^2+(y-5)^2)=y+1#

and squaring #(x-6)^2+(y-5)^2=(y+1)^2#

i.e. #x^2-12x+36+y^2-10y+25=y^2+2y+1#

i.e. #x^2-12x+60=12y#

or #12y=(x-6)^2+24#

or #y=1/12(x-6)^2+2#

graph{(x^2-12x+60-12y)((x-6)^2+(y-5)^2-0.03)(y+1)=0 [-4.08, 15.92, -1.32, 8.68]}