Chapter 10: Q. 44 (page 683)

Derive equation (c) in Table 5:
$r = \frac{e p}{1 + e \sin θ}$

Short Answer

Expert verified

The polar equation $r = \frac{e p}{1 + e \sin θ}$ has been proved.

Step by step solution

Step 1. Given information

We have to derive the following equation :

$r = \frac{e p}{1 + e \sin θ}$

Step 2. Examine the given polar equation

To do that, let us examine first the given polar equation. We can observed that the denominator has $a + \sin θ$ term which means that the directrix is parallel to the polar axis and is located p units above to the right of the pole.

Step 3. Find the value of D

Based from the figure,

$D = p - l$

Find the value of l,

$\sin θ = \frac{l}{r} \Rightarrow l = r \sin θ$

Substitute the value,

$D = p - r \sin θ$

Step 4. Derive the equation

Relation between radius and eccentricity is given by :

$r = e D$

Apply previous equation to the equation of r,

$\begin{array}{r} r = e (p - r \sin θ) \\ r \overset{(1)}{=} e p - e r \sin θ \\ r + e r \sin θ \overset{(2)}{=} e p \\ r (1 + e \sin θ) \overset{(3)}{=} e p \\ r \overset{(4)}{=} \frac{e p}{1 + e \sin θ} \end{array}$

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Derive equation (c) in Table 5:
$r = \frac{e p}{1 + e \sin θ}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Examine the given polar equation

Step 3. Find the value of D

Step 4. Derive the equation

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Derive equation (c) in Table 5:r=ep1+esin⁡θ

Short Answer

Step by step solution

Step 1. Given information

Step 2. Examine the given polar equation

Step 3. Find the value of D

Step 4. Derive the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Pure Maths

Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Derive equation (c) in Table 5:
$r = \frac{e p}{1 + e \sin θ}$