What is the equation of the parabola with a focus at (-1,-4) and a directrix of y= -7?

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Answer 1 · 2016-07-26T11:57:23

#6y=x^2+2x-32#.

Let the Focus be #S(-1,-4)# and, let the Directrix be # d : y+7=0#.

By the Focus-Directrix Property of Parabola, we know that, for any pt. #P(x,y)# on the Parabola,

#SP= bot# Distance #D# from P to line #d#.

#:. SP^2=D^2#.

#:. (x+1)^2+(y+4)^2=|y+7|^2#

#:. x^2+2x+1=(y+7)^2-(y+4)^2#

#=(y+7+y+4)(y+7-y-4)=(2y+11)(3)=6y+33#

Hence, the Eqn. of the Parabola is given by,

#6y=x^2+2x-32#.

Recall that the formula to find the #bot# distance from a pt.#(h,k)# to a line #ax+by+c=0# is given by #|ah+bk+c|/sqrt(a^2+b^2)#.