What it does
Satellite dishes, solar cookers, torches and car headlights all use the same trick: a mirror shaped like a parabola. This page shows why.
- Rays come in parallel to the axis (orange), as light from a far-away satellite or the Sun does. Each one bounces off the dish by the law of reflection (blue) and every one passes through the focus F, however far from the centre it hits.
- Switch to Circle to compare a mirror shaped like part of a circle with the same focus for rays near the axis. Rays near the rim now cross the axis below F (red): a circle mirror only focuses well when it is shallow. This is called spherical aberration.
- Tick Light starts at the focus to run it backwards: a bulb at F sends light out as a straight, parallel beam.
- The numbers card gives the equation x² = 4ay (the textbook form y² = 4ax turned on its side), the focus, the directrix, the latus rectum and the depth of the dish, and explains the reflection with the focus–directrix property.
- Where does the receiver go? Measure a real dish across the rim (D) and its depth at the centre (d): the focus is D²/(16 d) above the centre.
It follows the Class 11 chapter conic sections (the parabola) and links to Class 10 and 12 work on mirrors.
The other conics, and where they come from on a cone, are in conic explorer; ordinary curved mirrors and their images are in mirror ray diagram, and the tangent lines at work here are straight lines from line forms converter.
How to use it
- Pick Parabola, Circle or Both.
- Set a (the focal length, from the bottom of the dish to F) and w (half the width of the dish).
- Tick the torch box to send light out from F instead.
- Type the width and depth of a real dish to find where its receiver should sit.
A worked example
The page starts with a = 2 and w = 4, so the dish is x² = 8y, 8 wide.
- Focus F = (0, 2), directrix y = −2, latus rectum 4a = 8.
- Depth at the rim = 4²/8 = 2, so F is level with the rim.
- The outermost ray hits at (3.8, 1.805). Its distance to F is 3.805, and its height above the directrix is 1.805 + 2 = 3.805: the same, as for every point on a parabola.
- Switch to Both: on the circle of radius 2a = 4, the ray at 3.8 crosses the axis at about −2.41, which is 4.41 below F.
A 60 cm dish that is 6 cm deep has a = 60²/(16 × 6) = 3600/96 = 37.5 cm.
Tips
- A smaller a makes a deeper dish with the focus closer to it.
- A shallow circle mirror is almost a parabola near its centre, which is why school mirror formulas use small, near-axis rays.
Check yourself
Four fresh questions: the focus of x² = 4ay, the latus rectum of y² = 4ax, the focus of a real dish from its size, and the distance of a point on the parabola from the focus. Nothing is saved.
FAQ
Does the unit matter? No. Use centimetres, metres or grid squares; the answers come out in the same unit.
Is this from the NCERT book? It covers the same topics with fibub’s own examples. fibub is not part of NCERT.
Where the numbers come from
- Standard results on the parabola x² = 4ay (focus (0, a), directrix y = −a, latus rectum 4a, the reflection property) and on spherical mirrors (a ray at height h crosses the axis R − R/(2 cos θ) from the vertex, sin θ = h/R). Written for fibub.
- Checked with the law of reflection on 300 random rays (each parabola ray passes through the focus; circle-mirror crossings match the formula), and by hand: a 60 cm dish 6 cm deep has its focus 37.5 cm above the centre.
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Last updated 2026-11-09.