What is the equation in standard form of the parabola with a focus at (-2,3) and a directrix of y= -9?

1 Answer
Oct 30, 2017

y=(x^2)/24+x/6-17/6y=x224+x6176

Explanation:

Sketch the directrix and focus (point AA here) and sketch in the parabola.
Choose a general point on the parabola (called BB here).
Join ABAB and drop a vertical line from BB down to join the directrix at CC.
A horizontal line from AA to the line BDBD is also useful.
enter image source here
By the parabola definition, point BB is equidistant from the point AA and the directrix, so ABAB must equal BCBC.
Find expressions for the distances ADAD, BDBD and BCBC in terms of xx or yy.
AD=x+2AD=x+2
BD=y-3BD=y3
BC=y+9BC=y+9
Then use Pythagoras to find AB:
AB=sqrt((x+2)^2+(y-3)^2)AB=(x+2)2+(y3)2
and since AB=BCAB=BC for this to be a parabola (and squaring for simplicity):
(x+2)^2+(y-3)^2=(y+9)^2(x+2)2+(y3)2=(y+9)2
This is your parabola equation.
If you want it in explicit y=... form, expand the brackets and simplify to give y=(x^2)/24+x/6-17/6