Reflection property of parabolas Consider the parabola y 4p with its focus at F(0, p) (see figure). The goal is to show that the angle of incidence between the ray € and the tangent line L (æ in the figure) equals the angle of reflection between the line PF and L (B in the figure). If these two angles are equal, then the reflec- tion property is proved because e is reflected through F. a. Let P(xp. Yo) be a point on the parabola. Show that the slope of the line tangent to the curve at Pis tan 0 P – Yo 2p b. Show that tan ọ = c. Show that a = 0; therefore, tan a = cot 0. 2 d. Note that B = 0 + p. Use the tangent addition tan 0 + tan o %3D formula tan (0 + p) to show that 1 - tan 0 tan o 2p tan a = tan B e. Conclude that because a and B are acute angles, a = B. F(0, p) Yo
Reflection property of parabolas Consider the parabola y 4p with its focus at F(0, p) (see figure). The goal is to show that the angle of incidence between the ray € and the tangent line L (æ in the figure) equals the angle of reflection between the line PF and L (B in the figure). If these two angles are equal, then the reflec- tion property is proved because e is reflected through F. a. Let P(xp. Yo) be a point on the parabola. Show that the slope of the line tangent to the curve at Pis tan 0 P – Yo 2p b. Show that tan ọ = c. Show that a = 0; therefore, tan a = cot 0. 2 d. Note that B = 0 + p. Use the tangent addition tan 0 + tan o %3D formula tan (0 + p) to show that 1 - tan 0 tan o 2p tan a = tan B e. Conclude that because a and B are acute angles, a = B. F(0, p) Yo
Chapter4: Rational Functions And Conics
Section: Chapter Questions
Problem 9PS
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Transcribed Image Text:Reflection property of parabolas Consider the parabola y
4p
with its focus at F(0, p) (see figure). The goal is to show that the
angle of incidence between the ray € and the tangent line L (æ in
the figure) equals the angle of reflection between the line PF and
L (B in the figure). If these two angles are equal, then the reflec-
tion property is proved because e is reflected through F.
a. Let P(xp. Yo) be a point on the parabola. Show that the slope of
the line tangent to the curve at Pis tan 0
P – Yo
2p
b. Show that tan ọ
=
c. Show that a =
0; therefore, tan a = cot 0.
2
d. Note that B = 0 + p. Use the tangent addition
tan 0 + tan o
%3D
formula tan (0 + p)
to show that
1 - tan 0 tan o
2p
tan a = tan B
e. Conclude that because a and B are acute angles, a = B.
F(0, p)
Yo
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