Why reflectors are parabolic.

Rays parallel to the axis hit y² = 4ax and reflect through the focus; dragging a lamp to the focus makes the outgoing beam parallel.

Move the lamp until every reflected ray leaves parallel to the axis (lamp at the focus).

• Most applied problems come down to choosing axes so that the curve takes a standard form, then reading one known point off the picture.
• A parabolic mirror with its focus 5 cm from the vertex and a depth of 45 cm: with the vertex at the origin and the axis along the x-axis, y² = 20x. At x = 45, y² = 900 and y = ±30, so the rim is 60 cm across.
• A beam resting on supports 12 m apart sags 3 cm at the centre in a parabola. With the lowest point as origin, x² = 4ay passes through (6, 3/100), so a = 300 m. The sag is 1 cm where the beam is 2 cm above the lowest point: x² = 4 × 300 × 2/100 = 24, that is 2√6 m from the centre.
• A rod AB of length 15 cm slides with A on the x-axis and B on the y-axis. The point P of the rod with AP = 6 cm satisfies cos θ = x/9 and sin θ = y/6, so x²/81 + y²/36 = 1: P moves on an ellipse.
• Satellite dishes, torch reflectors and vehicle headlights use the parabola: rays arriving parallel to the axis are reflected through the focus, and a source at the focus sends light out parallel to the axis.
• Two fixed points with a constant sum of distances (a running track round two flag posts, a string pinned at two points) give an ellipse; a constant difference gives a hyperbola.

Teaching this idea? Put the simulation on your own page or LMS with this code.