Chapter 10.6, Problem 17E

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8th Edition

James Stewart

ISBN: 9781305270336

Textbook Problem

- (a) Find the eccentricity and directrix of the conic
*r*= 1/(1 − 2 sin*θ*) and graph the conic and its directrix. - (b) If this conic is rotated counterclockwise about the origin through an angle 3
*π*/4, write the resulting equation and graph its curve.

To determine

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

Explanation

**Given:**

The given polar equation is

**Calculation:**

(a)

Calculate the eccentricity for the given equation.

The polar equation for the given equation will be either

Compare the given equation with the polar equation. Therefore,

Therefore, **the eccentricity for the given equation is**

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

Write the equation of directrix.

From the equation (1),

Substitute

Therefore, the equation of directrix is

(b)

The value of x and y with respect to â€˜râ€™ is,

Calculate the value of *r* for the various value of

Substitute

Calculate the value of *x*.

Substitute
*r* in the above equation.

Calculate the value of *y*.

Substitute
*r* in the above equation

Similarly calculate the remaining values.

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

0 | 1.00 | 1.00 | 0.00 |

10 | 1.53 | 1.51 | 0.27 |

20 | 3.16 | 2.97 | 1.08 |

40 | -3.50 | -2.68 | -2.25 |

50 | -1.88 | -1.21 | -1.44 |

60 | -1.37 | -0.68 | -1.18 |

70 | -1.14 | -0.39 | -1.07 |

80 | -1.03 | -0.18 | -1.02 |

90 | -1.00 | 0.00 | -1.00 |

100 | -1.03 | 0.18 | -1.02 |

110 | -1.14 | 0.39 | -1.07 |

120 | -1.37 | 0.68 | -1.18 |

130 | -1.88 | 1.21 | -1.44 |

140 | -3 |

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