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).
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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) and directrix is y=3. We start by assuming a general point on the parabola (x,y).
Using the distance formula, we find that the distance between (x,y) and the focus (−2,5) is (x+2)2+(y−5)2, and the distance between (x,y) and the directrix y=3 is (y−3)2. On the parabola, these distances are equal: