   Chapter 10, Problem 20RE

Chapter
Section
Textbook Problem

# Graph the ellipse r = 2/(4 − 3 cos θ) and its directrix. Also graph the ellipse obtained by rotation about the origin through an angle 2π/3.

To determine

To find: The ellipse curve and its directrix for the given polar equation and also plot the ellipse curve by rotating through an angle 120 degree.

Explanation

Given:

The polar equation is r=243cosθ .

Calculation:

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 4.

r=243cosθr=244434cosθr=12134cosθ

Therefore, the eccentricity for the given equation is 34 .

Write the equation of directrix.

ed=12 (1)

Substitute 34 for e in the equation (1).

ed=1234×d=12d=12×43d=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=243cosθ

Substitute 0° for θ in the above equation.

r=243cos(0°×π180)=2.0

Calculate the value of x.

x=rcosθ

Substitute the values of 2.0 for r and 0° for θ in the above equation.

x=2.0×cos(0°×π180)=2.0

Calculate the value of y.

y=rsinθ

Substitute the values of 2.0 for r and 0° for θ in the above equation.

y=2.0×sin(0°×π180)=0.0

Similarly, calculate the remaining values.

Tabulate the values as shown in table (1).

 θ r=24−3cosθ x=rcosθ y=rsinθ 0.00 2.00 2.00 0.00 10.00 1.91 1.88 0.33 20.00 1.69 1.59 0.58 30.00 1.43 1.24 0.71 40.00 1.18 0.90 0.76 50.00 0.97 0.62 0.74 60.00 0.80 0.40 0.69 70

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