Chapter 9: Q. 17 (page 772)

Let $α, β$ , and $γ$ be nonzero constants. Show that the graph of $r = \frac{α}{β + γ \sin θ}$ is a conic section with eccentricity $|\frac{γ}{β}|$ and directrix $y = \frac{α}{γ}$ .

Short Answer

Expert verified

Ans: The eccentricity is $|\frac{γ}{β}|$ and the directrix is $y = \frac{α}{γ}$ .

Step by step solution

Step 1. Given information:

the non zero constants $α, β$ and $γ$ of the equation $r = \frac{α}{β + γ \sin θ}$ .

Step 2. Converting the equation to the standard form:

The equation $r = \frac{α}{β + γ \sin θ}$ is not in the standard form of the polar equation

$r = \frac{e u}{1 + e \sin θ}$

Convert the equation to the standard form by dividing with $β$ in the numerator and in the denominator.

Then,

$r = \frac{\frac{α}{β}}{\frac{β + γ \sin θ}{β}}$ since dividing by $β$

$r = \frac{\frac{α}{β}}{\frac{β}{β} + \frac{γ \sin θ}{β}}$

$r = \frac{\frac{α}{β}}{1 + \frac{γ}{β} \cdot \sin θ}$

Step 3. multiplying and dividing the numerator:

Now multiply and divide by $γ$ in the numerator to make it in the standard form.

$r = \frac{\frac{γ}{γ} \cdot \frac{α}{β}}{1 + \frac{γ}{β} \cdot \sin θ}$

$r = \frac{\frac{γ}{β} \cdot \frac{α}{γ}}{1 + \frac{γ}{β} \cdot \sin θ}$

The equation $r = \frac{\frac{γ}{β} \cdot \frac{α}{γ}}{1 + \frac{γ}{β} \cdot \sin θ}$ is in the standard form.

Step 4. Comparing the equation for finding the eccentricity:

Now compare the equation $r = \frac{\frac{γ}{β} \cdot \frac{α}{γ}}{1 + \frac{γ}{β} \cdot \sin θ}$ with $r = \frac{e u}{1 + e \sin θ}$ .

For the polar equation $r = \frac{\frac{γ}{β} \cdot \frac{α}{γ}}{1 + \frac{γ}{β} \cdot \sin θ}$ the eccentricity is equals to $\frac{γ}{β}$ which is taken as positive .so eccentricity is $|\frac{γ}{β}|$ and the directrix is $y = \frac{α}{γ}$ .

Hence it is proved.

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Let $α, β$ , and $γ$ be nonzero constants. Show that the graph of $r = \frac{α}{β + γ \sin θ}$ is a conic section with eccentricity $|\frac{γ}{β}|$ and directrix $y = \frac{α}{γ}$ .

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Converting the equation to the standard form:

Step 3. multiplying and dividing the numerator:

Step 4. Comparing the equation for finding the eccentricity:

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Let α,β, and γ be nonzero constants. Show that the graph of r=αβ+γsinθ is a conic section with eccentricity γβ and directrix y=αγ.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Converting the equation to the standard form:

Step 3. multiplying and dividing the numerator:

Step 4. Comparing the equation for finding the eccentricity:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Geometry

Calculus

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Let $α, β$ , and $γ$ be nonzero constants. Show that the graph of $r = \frac{α}{β + γ \sin θ}$ is a conic section with eccentricity $|\frac{γ}{β}|$ and directrix $y = \frac{α}{γ}$ .