Chapter 10 Group Activity: Focal Chords
Chapter 10
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Group Activity
Focal Chords
Many of the applications of the conic sections are based on their reflective or
focal properties. One of the interesting algebraic properties of the conic sections
concerns their focal chords.
If a line through a focus F contains two points G and H of a conic section,
then the line segment GH is called a focal chord. Let G(x1, y1) and H(x2, y2) be
points on the graph of x2 ⫽ 4ay such that GH is a focal chord. Let u denote the
length of GF and v the length of FH (see Fig. 1).
FIGURE 1
y
Focal chord GH of the parabola
x2 ⫽ 4ay.
G
F
u
v
H
(2a, a)
x
(A) Use the distance formula to show that u ⫽ y1 ⫹ a.
(B) Show that G and H lie on the line y ⫺ a ⫽ mx, where m ⫽ (y2 ⫺ y1)/
(x2 ⫺ x1).
(C) Solve y ⫺ a ⫽ mx for x and substitute in x2 ⫽ 4ay, obtaining a quadratic
equation in y. Explain why y1y2 ⫽ a2.
1 1 1
(D) Show that ⫹ ⫽ .
u v a
(u ⫺ 2a)2
(E) Show that u ⫹ v ⫺ 4a ⫽
. Explain why this implies that
u⫺a
u ⫹ v ⱖ 4a, with equality if and only if u ⫽ v ⫽ 2a.
(F) Which focal chord is the shortest? Is there a longest focal chord?
1 1
(G) Is ⫹ a constant for focal chords of the ellipse? For focal chords of the
u v
hyperbola? Obtain evidence for your answers by considering specific
examples.
(H) The conic section with focus at the origin, directrix the line x ⫽ D ⬎ 0, and
DE
eccentricity E ⬎ 0 has the polar equation r ⫽
. Explain how this
1 ⫹ E cos ␪
1 1 1
polar equation makes it easy to show that ⫹ ⫽ for a parabola. Use
u v a
1 1
the polar equation to determine the sum ⫹ for a focal chord of an ellipse
u v
or hyperbola.