Explanation Calculation: From the given Circle, it is clear that, the graph is symmetrical to only y -axes and it is not symmetrical to x -axis and the origin. Therefore, the probable equation for the circle from the given equations is r = − sin θ . Verify whether the polar equation r = − sin θ satisfy the symmetrical status of circle or not as follows: Condition for symmetric about the x -axis: If the point ( r , θ ) lies on a curve, then the point ( r , − θ ) or ( − r , π − θ ) also lies on the curve in order to satisfy the condition for symmetric about the x -axis. Condition for symmetric about the y -axis: If the point ( r , θ ) lies on a curve, then the point ( r , π − θ ) or ( − r , − θ ) also lies on the curve in order to satisfy the condition for symmetric about the y -axis. Condition for symmetric about the origin: If the point ( r , θ ) lies on a curve, then the point ( − r , θ ) or ( − r , θ + π ) also lies on the curve in order to satisfy the condition for symmetric about the origin. Rewrite the probable polar coordinate equation as follows: r = − sin θ (1) Substitute ( r , θ ) in the Equation (1) as follows: r = − sin θ (2) Substitute ( r , − θ ) in the Equation (1) as follows: r<

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Chapter 11, Problem 43PE

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Identify the appropriate equation for the given Circle from the given equations.

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Chapter 11, Problem 42PE

Chapter 11, Problem 44PE

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