College Algebra 9th txtbk

422 CHAPTER 6 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
CHAPTER 6
ZZZ 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 (x 1 , y 1 ) and H (x 2 , y 2 ) be points on the graph of
x 2 4ay such that GH is a focal chord. Let u denote the length of
GF and v the length of FH (Fig. 1).
G
F
u
y
H
(2a, a)
x
Z Figure 1 Focal chord GH of the
parabola x 2 4ay.
v
(A) Use the distance formula to show that u y 1 a.
(B) Show that G and H lie on the line y a mx, where
m (y 2 y 1 )(x 2 x 1 ).
(C) Solve y a mx for x and substitute in x 2 4ay, obtaining
a quadratic equation in y. Explain why y 1 y 2 a 2 .
1
(D) Show that
u 1 v 1 a .
(u 2a)2
(E) Show that u v 4a Explain why this
u a .
implies that u v 4a, with equality if and only if u v 2a.
(F) Which focal chord is the shortest? Is there a longest focal
chord?
1
(G) Is a constant for focal chords of the ellipse? For
u 1 v
focal chords of the hyperbola? Obtain evidence for your
answers by considering specific examples.