Conics in the real world

Conics are everywhere: dish aerials and torch mirrors are parabolas, arches and planet orbits are ellipses, and cooling towers have hyperbolic sides. The method for a word problem is always the same:

• Choose the axes so the curve is in a standard position: vertex at the origin for a parabola, centre at the origin for an ellipse.
• Write the standard equation with one or two unknowns.
• Put in a point you know, solve, then answer the question.

A dish (parabolic reflector) 24 cm across and 6 cm deep. Where is the focus?

• Vertex at the origin, axis along the x-axis: y² = 4ax.
• The rim is 6 cm along the axis and 12 cm to each side, so (6, 12) is on it: 144 = 24a, so a = 6.
• The focus is 6 cm from the vertex, along the axis. Signals hitting the dish all bounce to that point, which is why the receiver goes there.

A semi-elliptical arch 10 m wide at the base and 4 m high in the middle. How high is it 3 m from the centre?

• Centre of the base at the origin: x²/25 + y²/16 = 1 (a = 5, b = 4).
• At x = 3: y² = 16(1 – 9/25) = 16 × 16/25, so y = 3.2 m.

A hanging cable shaped like a parabola, between towers 100 m apart, sagging 20 m below the tower tops at its lowest point. How high above the lowest point is the cable 20 m from the middle?

• Lowest point at the origin: x² = 4ay. A tower top is at (50, 20), so 2500 = 80a and the equation is x² = 125y.
• At x = 20: y = 400/125 = 3.2 m.

Sums of distances make ellipses. Someone moves so that their distances from two posts 6 m apart always add up to 10 m. Then 2a = 10 and 2c = 6, so a = 5, c = 3, b = 4: the path is x²/25 + y²/16 = 1.

A sliding rod. A 12 m rod slides with one end on the x-axis and the other on the y-axis. Take the point P that is 4 m from the end on the x-axis. If the rod makes angle θ with the x-axis, P is at (8 cos θ, 4 sin θ).

• So (x/8)² + (y/4)² = cos²θ + sin²θ = 1.
• P moves on the ellipse x²/64 + y²/16 = 1, not on a circle (only the mid-point of the rod traces a circle).