Show that the ellipse x 2 / a 2 + y 2 / b 2 = 1 and the hyperbola x 2 / A 2 — y 2 / B 2 = 1 are orthogonal trajectories if A 2 < a 2 and a 2 — b 2 = A 2 + B 2 (so the ellipse and hyperbola have the same foci).
Solution Summary: The author explains that the ellipse and the hyperbola are orthogonal trajectories of each other if the product of slopes of tangents to the curves at points of intersection equal
Show that the ellipse
x
2
/
a
2
+
y
2
/
b
2
=
1
and the hyperbola
x
2
/
A
2
—
y
2
/
B
2
=
1
are orthogonal trajectories if
A
2
<
a
2 and
a
2
—
b
2
=
A
2
+
B
2
(so the ellipse and hyperbola have the same foci).
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