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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

Graph the conics r = e/(1 − e cos θ) with e = 0.4, 0.6, 0.8, and 1.0 on a common screen. How does the value of e affect the shape of the curve?

To determine

To find: The variation of shape of curve with respect to eccentricity provided.

Explanation

Given:

The given polar equation is r=e1ecosθ with e=0.4,0.6,0.8and1.0.

Calculation:

Write the condition of conic:

  • If eccentricity is less than one will make conic of Ellipse
  • If eccentricity is equal to one will make conic of Parabola
  • If eccentricity is greater than one will make conic of Hyperbola

Write the polar equation with different eccentricities.

r=0.410.4cosθEllipser=0.610.6cosθEllipser=0.810.8cosθEllipser=111cosθParabola

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

x=rcosθy=rsinθ

Consider the eccentricity of 0.4.

Calculate the value of r.

r=0.410.4cosθ

Substitute 0° for θ in the above equation.

r=0.410.4cos(0°×π180)=0.67

Calculate the value of x.

x=rcosθ

Substitute 0.67 for r and 0° for θ in the above equation

x=0.67×cos(0°×π180)=0.67

Calculate the value of y.

y=rsinθ

Substitute 0.67 for r and 0° for θ in the above equation

y=0.67×sin(0°×π180)=0

Similarly calculate the remaining values.

Tabulate the calculated value as shown in the table (1).

θr=0.410.4cosθx=rcosθy=rsinθ
00.670.670.00
100.660.650.11
200.640.600.22
300.610.530.31
400.580.440.37
500.540.350.41
600.500.250.43
700.460.160.44
800.430.070.42
900.400.000.40
1000.37-0.060.37
1100.35-0.120.33
1200.33-0.170.29
1300.32-0.200.24
1400.31-0.230.20
1500.30-0.260.15
1600.29-0.270.10
1700.29-0.280.05
1800.29-0.290.00

Similarly calculate the remaining values till 360 degrees.

Consider the eccentricity of 0.6.

Calculate the value of r.

r=0.610.6cosθ

Substitute 0° for θ in the above equation.

r=0.610.6cos(0°×π180)=1.5

Calculate the value of x.

x=rcosθ

Substitute 1.5 for r and 0° for θ in the above equation

x=1.5×cos(0°×π180)x=1.5

Calculate the value of y.

y=rsinθ

Substitute 1.5 for r and 0° for θ in the above equation

y=1.5×sin(0°×π180)y=0

Similarly calculate the remaining values.

Tabulate the calculated value as shown in the table (2).

θr=0.610.6cosθx=rcosθy=rsinθ
01.501.500.00
101.471.440.25
201.381.290.47
301.251.080.62
401.110.850.71
500.980.630.75
600.860.430.74
700.750.260.71
800.670.120.66
900.600.000.60
1000.54-0.090.54
1100.50-0.170.47
1200.46-0.230.40
1300.43-0.280.33
1400.41-0.310

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