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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Show that the angles between the polar axis and the asymptotes of the hyperbola r = ed/(1 − e cos θ), e > 1, are given by cos−1(±1/e).

To determine

To find: show that the condition r=ed1ecosθ is true for all value of eccentricity e>1 as given by cos1(±1e) .

Explanation

Given:

Calculation:

For a conic with the eccentricity value of e greater than 1, then it is a hyperbola,

a2=e2d2(1e2)2 (1)

b2=e2d21e2 (2)

When the condition e>1 is true, then the expression 1e2<0 is true.

Rewrite the equation (1) and (2) as follows.

a2=e2d2(e21)2 (3)

b2=e2d2e21 (4)

Divide equation (4) by (3).

b2a2=e2d2e21e2d2(e21)2=(e21)2e21

b2a2=e21(ba)2=e21

(ba)=±e21

Equation for asymptote is given as y=±bax

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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