What is the equation of the parabola with a focus at (7,5) and a directrix of y= -3?

1 Answer
Shwetank Mauria
Mar 30, 2018

Parabola's equation is #y=1/16(x-7)^2+1# and vertex is #(7,1)#.

Explanation:

Parabola is locus of a point which moves so that its distance from a given point calld focus and a given line ccalled directrix is always constant.

Let the point be #(x,y)#. Here focus is #(7,5)# and distance from focus is #sqrt((x-7)^2+(y-5)^2)#. Its distance from directrix #y=-3# i.e. #y+3=0# is #|y+3|#.

Hence equaion of parabola is

#(x-7)^2+(y-5)^2)=|y+3|^2#

or #x^2-14x+49+y^2-10y+25=y^2+6y+9#

or #x^2-14x+65=16y#

i.e. #y=1/16(x^2-14x+49-49)+65/16#

or #y=1/16(x-7)^2+(65-49)/16#

or #y=1/16(x-7)^2+1#

Hence parabola's equation is #y=1/16(x-7)^2+1# and vertex is #(7,1)#.

graph{(1/16(x-7)^2+1-y)((x-7)^2+(y-1)^2-0.15)((x-7)^2+(y-5)^2-0.15)(y+3)=0 [-12.08, 27.92, -7.36, 12.64]}

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