Key takeaway: The focus and directrix of a parabola are a fixed point and a fixed line, and the curve consists of every point that sits the same distance from both.
Grade 10 – GeometryG-GPE.A.2Geometry and Measurement
Definition of Focus and Directrix
The focus and directrix give a parabola a definition with no formula at all. Pick a point, the focus, and a line, the directrix. Every spot on the curve is as far from the point as from the line.
The two sit on opposite sides of the vertex, each at the same distance \(p\) from it. The curve bends away from the line and hugs the point. Satellite dishes and headlights exploit this: signals arriving parallel to the axis bounce to the focus, and a bulb placed there sends light out in parallel beams.
Formula and notation
For a parabola opening upward with its vertex at the origin, the focus and directrix are the point \((0, p)\) and the line \(y = -p\):
\[ x^2 = 4py \]
It comes from setting the distance from \((x, y)\) to \((0, p)\) equal to the distance from \((x, y)\) to the line.
Key facts
- The vertex lies exactly midway between the point and the line.
- The line is perpendicular to the axis of symmetry.
- A bigger p means a wider, flatter curve.
- Rays arriving parallel to the axis all reflect through the point.
Focus and Directrix example
A reflector has its bulb at \((0, 1.5)\) and a guide line \(y = -1.5\), in inches. Write the equation of its parabolic cross-section, and check that the point \((6, 6)\) belongs to it.
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The vertex is halfway between, at the origin, so \(p = 1.5\) and \(x^2 = 4(1.5)y = 6y\).
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Test \((6, 6)\): \(6^2 = 36\) and \(6y = 36\), so the point fits.
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Distance to the bulb: \(\sqrt{6^2 + (6-1.5)^2} = \sqrt{56.25} = 7.5\). Distance to the line: \(6-(-1.5) = 7.5\).
The cross-section is \(x^2 = 6y\), and \((6, 6)\) is 7.5 in. from both the bulb and the guide line.
Common mistake
Watch out
Lena says the bulb sits on the curve, since it belongs to the definition of the parabola.
Why it is wrong
The bulb sits inside the bend, not on the curve. Its distance to the guide line is \(2p = 3\), and a point of the curve must be equally far from both. The vertex, 1.5 from each, is the closest point of the curve to the bulb.
Related terms and lessons
- Geometry · Parabolas from Focus and Directrix, and Tangent Lines to a CircleLessonWorksheetQuiz
- Precalculus · Conic Sections: Parabolas, Ellipses and HyperbolasLessonWorksheetQuiz
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