The accompanying figure shows a typical point P(x 0 , y 0 ) on the parabola y
Question:
The accompanying figure shows a typical point P(x0, y0) on the parabola y2 = 4px. The line L is tangent to the parabola at P. The parabola’s focus lies at F( p, 0). The ray L′ extending from P to the right is parallel to the x-axis. We show that light from F to P will be reflected out along L′ by showing that β equals α. Establish this equality by taking the following steps.
a. Show that tan β = 2p/y0.
b. Show that tan ϕ = y0/(x0 - p).
c. Use the identity
to show that tan a = 2p/y0.
Since a and b are both acute, tan β = tan a implies β = α.
This reflective property of parabolas is used in applications like car headlights, radio telescopes, and satellite TV dishes.
Step by Step Answer:
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir