SOLUTION: Find the equation of r = 1/(2 + sinθ) in rectangular form.Show the step-by-step process.

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Question 972005: Find the equation of r = 1/(2 + sinθ) in rectangular form.Show the step-by-step process.
Answer by stanbon(75887) About Me  (Show Source):
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Find the equation of r = 1/(2 + sinθ) in rectangular form.Show the step-by-step process.
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Conversion forms:
r = sqrt(x^2+y^2)
theta = arctan(y/x)
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Your problem::
sqrt(x^2+y^2) = 1/(2 + (y/sqrt(x^2+y^2))
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Cross-multiply to get:
2sqrt(x^2+y^2) + y = 1
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2sqrt(x^2+y^2) = 1-y
Square both sides to get:
4(x^2+y^2) = 1 - 2y + y^2
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4x^2 + 4y^2 = 1 - 2y + y^2
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1-2y = 4x^2 + 3y^2
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3y^2 + 2y + 4x^2 = 1
Complete the square to get:
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3(y^2+(2/3)y + (1/9) + 4x^2 = 1+ 3(1/9)
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3(y+(1/3))^2 + 4x^2 = (4/3)
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(y+(1/3))^2/(4/9) + 4x^2/(4/3) = 1
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Cheers,
Stan H.
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