What is the standard form of the equation of the parabola with a directrix at x=23 and a focus at (5,5)?

1 Answer

The equation of parabola will be: #(y-5)^2=-36(x-14)#

Explanation:

Given equation of directrix of parabola is #x=23# & the focus at #(5, 5)#. It is clear that it is a horizontal parabola with sides diverging in -ve x-direction. Let general equation of parabola be
#(y-y_1)^2=-4a(x-x_1)# having equation of directrix: #x=x_1+a# & the focus at #(x_1-a, y_1)#
Now, comparing with given data, we have #x_1+a=23#, #x_1-a=5, y_1=5# which gives us #x_1=14, a=9# hence the equation of parabola will
#(y-5)^2=-4\cdot 9(x-14)#
#(y-5)^2=-36(x-14)#