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## Algebra (all content)

### Unit 17: Lesson 7

Focus and directrix of a parabola

# Parabola focus & directrix review

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).
A parabola that opens up. Above the vertex of the parabola is a point labeled focus. Below the parabola is a horizontal line labeled directirix. On the parabola, there are three points at random locations. Each point has a line segment that attaches to the focus and a line segment that attaches to the directrix. The pairs of segments that leave a point on the parabola have equal distances.

## 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 left parenthesis, minus, 2, comma, 5, right parenthesis and directrix is y, equals, 3. We start by assuming a general point on the parabola left parenthesis, x, comma, y, right parenthesis.
Using the distance formula, we find that the distance between left parenthesis, x, comma, y, right parenthesis and the focus left parenthesis, minus, 2, comma, 5, right parenthesis is square root of, left parenthesis, x, plus, 2, right parenthesis, squared, plus, left parenthesis, y, minus, 5, right parenthesis, squared, end square root, and the distance between left parenthesis, x, comma, y, right parenthesis and the directrix y, equals, 3 is square root of, left parenthesis, y, minus, 3, right parenthesis, squared, end square root. On the parabola, these distances are equal:
\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}