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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

Expert verified

Ans: The eccentricity is γβand the directrix is y=αγ.

Step by step solution

01

Step 1. Given information: 

the non zero constants α,β and γ of the equation r=αβ+γsinθ.

02

Step 2. Converting the equation to the standard form:

The equation r=αβ+γsinθis not in the standard form of the polar equation

r=eu1+esinθ

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

Then,

r=αββ+γsinθβsince dividing by β

r=αβββ+γsinθβ

r=αβ1+γβ·sinθ

03

Step 3. multiplying and dividing the numerator:

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

r=γγ·αβ1+γβ·sinθ

r=γβ·αγ1+γβ·sinθ

The equation r=γβ·αγ1+γβ·sinθis in the standard form.

04

Step 4. Comparing the equation for finding the eccentricity: 

Now compare the equation r=γβ·αγ1+γβ·sinθwith r=eu1+esinθ.

For the polar equation r=γβ·αγ1+γβ·sinθthe eccentricity is equals to γβwhich is taken as positive .so eccentricity is γβand the directrix is y=αγ.

Hence it is proved.

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