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

4 Answers

  • 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!

  • if r = -4cosθ

    r^2 = -4rcosθ

    now r^2 = x^2 + y^2

    and x = rcosθ, so we have

    x^2 + y^2 = -4x

    thus

    x^2 + 4x + 4 + y^2 = 4

    or

    (x + 2)^2 + y^2 = 4

    which is a circle centered at (-2, 0) of radius 2.

  • r = -4cosθ

    r² = -4r cosθ

    Does that help? See if you can show that this is a circle, centre (-2, 0), radius 2.

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

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