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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Graph the conic r = 4/(5 + 6 cos θ) and its directrix. Also graph the conic obtained by rotating this curve about the origin through an angle π/3.

(a)

To determine

To find: The eccentricity, equation of directrix and draw the conic and its directrix

Explanation

Given:

The given polar equation is r=45+6cosθ (1)

Calculation:

(a)

Calculate the eccentricity for the given equation.

The polar equation for the given equation will be either r=ed1±ecosθ

To make the given equation to polar equation, divide the given equation by 5.

r=45+6cosθr=4555+65cosθr=451+65cosθ

Therefore, the eccentricity for the given equation is65.

Here, the eccentricity is greater than one, so the conic is Hyperbola.

Write the equation of directrix.

From the equation (1),

ed=45 (2)

Substitute 65 for e in the equation (2).

ed=4565×d=45d=45×56d=23

Therefore, the equation of directrix is x=23

The value of x and y with respect to ‘r’ is,

x=rcosθy=rsinθ

Calculate the value of r for the various value of θ

r=45+6cosθ

Substitute 0° for θ in the above equation.

r=45+6cos(0°×π180)=0.5

Calculate the value of x.

x=rcosθ

Substitute 0.5 for r and 0° for θ in the above equation

x=0.5×cos(0°×π180)=0

(b)

To determine

To write: The polar equation for the conic which is rotated with angle π3

r=45+6cos(θπ3)

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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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