Study Guide for Stewart's Multivariable Calculus, 8th
Study Guide for Stewart's Multivariable Calculus, 8th
8th Edition
ISBN: 9781305271845
Author: Stewart, James
Publisher: Brooks Cole
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Textbook Question
Chapter 10.6, Problem 1PT

The figure at the right shows one point P on a conic and the distances of P to the focus and the directrix of a conic. What type of conic is it?

  1. a) parabola
  2. b) ellipse
  3. c) hyperbola
  4. d) not enough information is provided to answer the question.

Chapter 10.6, Problem 1PT, The figure at the right shows one point P on a conic and the distances of P to the focus and the

Expert Solution & Answer
Check Mark
To determine

To identify: The correct option for the blank in the statement, “The type of conic shown in the figure is a ____”.

Answer to Problem 1PT

The type of conic shown in the figure is a hyperbola which is option c.

Explanation of Solution

Given:

The options are, a) parabola, b) ellipse, c) hyperbola and d) not enough information is provided to answer the question.

Formula used:

Let PF be the distance between the point P on a conic and the focus F and, let Pl be the distance between the point on a conic and the directrix l. Then,

The conic is a parabola if the ratio |PF||Pl|=1.

The conic is an ellipse if the ratio |PF||Pl|=e<1.

The conic is a hyperbola if the ratio |PF||Pl|=e>1.

Calculation:

From the given figure, it is observed that the distance between the point P on a conic and the focus, F is 6 and the distance between the point on a conic and the directrix l is 4.

Compute the value of the ratio |PF||Pl| as follows.

|PF||Pl|=64=32=1.5>1

Therefore, the type of conic shown in the figure is a hyperbola which is option c.

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