CCSS Math: HSG.GPE.A.2
Review your knowledge of the focus and directrix of parabolas.

What are the focus and directrix of a parabola?

Parabolas are commonly known as the graphs of quadratic functions. They can also be viewed as the set of all points whose distance from a certain point (the focus) is equal to their distance from a certain line (the directrix).
Want to learn more about focus and directrix of a parabola? Check out this video.

Parabola equation from focus and directrix

Given the focus and the directrix of a parabola, we can find the parabola's equation. Consider, for example, the parabola whose focus is at (2,5)(-2,5) and directrix is y=3y=3. We start by assuming a general point on the parabola (x,y)(x,y).
Using the distance formula, we find that the distance between (x,y)(x,y) and the focus (2,5)(-2,5) is (x+2)2+(y5)2\sqrt{(x+2)^2+(y-5)^2}, and the distance between (x,y)(x,y) and the directrix y=3y=3 is (y3)2\sqrt{(y-3)^2}. On the parabola, these distances are equal:
(y3)2=(x+2)2+(y5)2(y3)2=(x+2)2+(y5)2y26y+9=(x+2)2+y210y+256y+10y=(x+2)2+2594y=(x+2)2+16y=(x+2)24+4\begin{aligned} \sqrt{(y-3)^2} &= \sqrt{(x+2)^2+(y-5)^2} \\\\ (y-3)^2 &= (x+2)^2+(y-5)^2 \\\\ \blueD{y^2}-6y\goldD{+9} &= (x+2)^2\blueD{+y^2}\maroonD{-10y}+25 \\\\ -6y\maroonC{+10y}&=(x+2)^2+25\goldD{-9} \\\\ 4y&=(x+2)^2+16 \\\\ y&=\dfrac{(x+2)^2}{4}+4\end{aligned}
Want to learn more about finding parabola equation from focus and directrix? Check out this video.

Check your understanding

Problem 1
Write the equation for a parabola with a focus at (6,4)(6,-4) and a directrix at y=7y=-7.
y=y=
Want to try more problems like this? Check out this exercise.
Loading