Convert the equation r cosine theta equals 9 sine (2 theta )to Cartesian coordinates. Describe the resulting curves. Choose the correct equa

Question

Convert the equation r cosine theta equals 9 sine (2 theta )to Cartesian coordinates. Describe the resulting curves. Choose the correct equations below. A. (x minus 9 )squared plus y squared equals 9 squared and x equals 0 B. x squared plus y squared equals 9 squared and x equals 0 C. x squared plus (y minus 9 )squared equals 9 squared and x equals 0 Your answer is correct.D. (x minus 9 )squared plus (y minus 9 )squared equals 9 squared and x equals 0 Choose the best description of the curves described by this equation. A. a circle centered at (negative 9 comma 0 )with a radius of 9 and the y dash axis B. a circle centered at (0 comma 9 )with a radius of 9 and the y dash axis Your answer is correct.C. a circle centered at (0 comma negative 9 )with a radius of 9 and the y dash axis D. a circle centered at (9 comma 0 )with a radius of 9 and the y dash axis

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Lydia 17 hours 2021-09-15T20:55:47+00:00 1 Answer 0

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    2021-09-15T20:56:53+00:00

    Answer:

    The curve is a circle of radius 9 centered at the point (0,9) and the equation is x^2+(y-9)^2 = 81

    Step-by-step explanation:

    Proceed as follows:

     r\cos(\theta) = 9 \sin(2\theta)

    Take \sin(2\theta) = 2\sin(\theta)\cos(\theta). Then

     r\cos(\theta) = 9 \sin(2\theta)= 18 \sin(\theta) \cos(\theta)

    Multiply both side by r^2. Then

     r^2\cdot r\cos(\theta) = 18 \cdot r\sin(\theta) \cdot r\cos(\theta)

    Use the following substitution  x = r\cos(\theta), y = r\sin(\theta), r^2 = x^2+y^2. Then

     (x^2+y^2)\cdot (x) = 18 \cdot (x)\cdot(y)

    By cancelling out x on both sides we get the following equation

    x^2+y^2 = 18y or x^2+y^2-18y =0

    Recall that given a expression of the form x^2-bx we can complete the square by adding an substracting the amount \frac{b^2}{4}. So, we get  x^2-bx = (x-\frac{b}{2})^2-\frac{b^2}{4}. In our case, we will complete the square for y, then

     x^2+y^2-18y = x^2+y^2-18y+(\frac{18}{2})^2-(\frac{18}{2})^2 = 0. Then

     x^2+(y-9)^2-81=0 or

    x^2+(y-9)^2 = 81.

    Recall that the equation of a circle is given by (x-h)^2+(y-k)^2 = r^2 where (h,k) is the center of the circle and r is the radius. In our case we have h=0, k = 9 and r = 9. So it is a circle of radius 9 centered at the point (0,9)

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