Resolved: "Consider the following three equations: A)r=63+2cos⁡(θ)"

Agaiepsh

Answered question

2021-11-15

Consider the following three equations:

A) r = \frac{6}{3 + 2 \cos (θ)}

Troy Lesure

Beginner2021-11-16Added 26 answers

Step 1
Solution:

A) r = \frac{6}{3 + 2 \cos θ}

\Rightarrow r (3 + 2 \cos θ) = 6

3 r + 2 r \cos θ = 6

3 \sqrt{x^{2} + y^{2}} + 2 x = 6

\Rightarrow 3 \sqrt{x^{2} + y^{2}} = 6 - 2 x

Squaring both the sides

9 (x^{2} + y^{2}) = 36 - 24 x + 4 x^{2}

9 x^{2} - 4 x^{2} - 24 x + 9 y^{2} = 36

5 x^{2} - 24 x + 9 y^{2} = 36

5 (x^{2} - \frac{24}{5} x) + 9 y^{2} = 36

5 (x^{2} - 2 \times \frac{12}{5} x + {(\frac{12}{5})}^{2}) + 9 y^{2} = 36 + 5 {(\frac{12}{5})}^{2}

5 {(x - \frac{12}{5})}^{2} + 9 y^{2} = 36 + \frac{144}{5}

5 {(x - \frac{12}{5})}^{2} + 9 y^{2} = \frac{324}{5}

Dividing throughout by

\frac{324}{5}

\frac{5 {(x - \frac{12}{5})}^{2}}{\frac{324}{5}} + \frac{9 y^{2}}{\frac{324}{5}} = 1

\Rightarrow \frac{{(x - \frac{12}{5})}^{2}}{\frac{324}{25}} + \frac{y^{2}}{\frac{324}{45}} = 1

\frac{{(x - \frac{12}{5})}^{2}}{{(\frac{18}{5})}^{2}} + \frac{y^{2}}{{(\frac{18}{\sqrt{45}})}^{2}} = 1

Hence, this is the equation of an ellipse of the form

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