Resolved: "Replace the Cartesian equation with equivalent polar equations...."

Ernstfalld

Answered question

2021-09-22

Replace the Cartesian equation with equivalent polar equations.

{(x + 2)}^{2} + {(y - 5)}^{2} = 16

Nicole Conner

Skilled2021-09-23Added 97 answers

Step 1
Given:
The Cartesian equation:

{(x + 2)}^{2} + {(y - 5)}^{2} = 16

Step 2
To convert into the polar equation, put

x = r \cos 0

y = r \sin 0

Therefore,

{(x + 2)}^{2} + {(y - 5)}^{2} = 16

{(r \cos 0 + 2)}^{2} + {(r \sin 0 - 5)}^{2} = 16

Expand using:

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

{(a + b)}^{2} = a^{2} + 2 a b + b^{2}

\cos^{2} 0 + \sin^{2} 0 = 1

(r^{2} \cos^{2} 0 + 2 \times r \cos 0 \times 2 + 2^{2}) + (r^{2} \sin^{2} 0 - 2 \times r \sin 0 \times 5 + 5^{2}) = 16

r^{2} \cos^{2} 0 + 4 r \cos 0 + 4 + r^{2} \sin^{2} 0 - 10 r \sin 0 + 25 = 16

r^{2} (\cos^{2} 0 + \sin^{2} 0) + 4 r \cos 0 - 10 r \sin 0 + 13 = 0

r^{2} + 4 r \cos 0 - 10 r \sin 0 + 13 = 0

Thus,
The equivalent polar equation is:

r^{2} + 4 r \cos 0 - 10 r \sin 0 + 13 = 0

Jeffrey Jordon

Expert2021-11-27Added 2605 answers

Answer is given below (on video)

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