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Mr. Ellis Hintz

Feb 20, 2025

Transform polar coordinates r = -4cosθ to an equation in rectangular coordinates?

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4 Answers

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Bernhard White

Feb 20, 2025

Use:r = √(x^2 + y^2) and cosθ = x/r.So we have: r = -4cosθ==> √(x^2 + y^2) = -4x/r, by substituting==> √(x^2 + y^2) = -4x/√(x^2 + y^2), since r = √(x^2 + y^2)==> x^2 + y^2 = -4x==> (x^2 + 4x) + y^2 = 0 ==> (x^2 + 4x + 4) + y^2 = 4, by completing the square==> (x + 2)^2 + y^2 = 4 = 2^2.This is a circle with a center of (-2, 0) and a radius of 2.I hope this helps!... Show More
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Bernhard White

Feb 20, 2025

if r = -4cosθr^2 = -4rcosθnow r^2 = x^2 + y^2and x = rcosθ, so we havex^2 + y^2 = -4xthusx^2 + 4x + 4 + y^2 = 4or(x + 2)^2 + y^2 = 4which is a circle centered at (-2, 0) of radius 2.... Show More
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Bernhard White

Feb 20, 2025

r = -4cosθr² = -4r cosθDoes that help? See if you can show that this is a circle, centre (-2, 0), radius 2.... Show More
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Bernhard White

Feb 20, 2025

rectangular coordinates (rcos θ, rsinθ) = (-4cos^2θ,-4sinθcosθ) = -2 {(cos2θ+1), sin(2θ)}... Show More
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