Constructing a Hyperbola In this problem we construct a hyperbola. The wooden bar in the figure can pivot at F1. A string that is shorter than the bar is anchored at F2 and at A, the other end of the bar. A pencil at P holds the string taut against the bar as it moves counterclockwise around F1.
- (a) Show that the curve traced out by the pencil is one branch of a hyperbola with foci at F1 and F2.
- (b) How should the apparatus be reconfigured to draw the other branch of the hyperbola?
(a)
To show: The curve traced out by the pencil is one branch of a hyperbola with foci at
Explanation of Solution
Definition used:
Geometric definition of hyperbola:
“A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points
From the given Figure, it is observed that wooden bar is pivot at
A pencil at P holds the string taut against the bar as it moves counterclockwise around
That is, the difference of the distances
Therefore, by the geometric definition of the hyperbola, the curve traced by the pencil is a hyperbola.
Since, P moves around
Hence, the curve traced out by the pencil is one branch of a hyperbola with foci at
(b)
How the apparatus to be reconfigured to draw the other branch of the hyperbola.
Explanation of Solution
Reconfigure the given apparatus as shown in Figure 1 to draw the other branch of the hyperbola.
From the given Figure, it is observed that wooden bar is pivot at
A pencil at P holds the string taut against the bar as it moves counterclockwise around
That is, the difference of the distances
Therefore, by the geometric definition of the hyperbola, the curve traced by the pencil is a hyperbola.
Since, P moves around
Chapter 11 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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