Q. 81

Question

Microphones In studiosand on stages, cardioid microphones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Suppose one such cardioid pattern is given by the equation ${(x^{2} + y^{2} - x)}^{2} = x^{2} + y^{2}$
.

(a) Find the intercepts of the graph of the equation.

(b) Test for symmetry with respect to the $x$ -axis, $y$ -axis, and origin.

Step-by-Step Solution

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Answer

The intercepts of the graph of the equation ${(x^{2} + y^{2} - x)}^{2} = x^{2} + y^{2}$ are:

$(0, 0), (2, 0) (0, 1) (0, - 1)$

The equation is symmetric with respect to the $x$ -axis,

1Step 1: Determine the x -intercept.

For $x$ -intercept, substitute $0$ for $y$ and then solve for $x$ .

$\begin{array}{rcl} {(x^{2} + 0^{2} - x)}^{2} & = & x^{2} + 0^{2} \\ {(x^{2} - x)}^{2} & = & x^{2} \\ x^{4} - 2 x^{3} + x^{2} & = & x^{2} \\ x^{4} - 2 x^{3} & = & 0 \\ x^{3} (x - 2) & = & 0 \\ x & = & 0, 2 \end{array}$

2Step 2: Determine the y -intercept.

For $y$ -intercept, substitute $0$ for $x$ and then solve for $y$ .

$\begin{array}{rcl} {(0^{2} + y^{2} - 0)}^{2} & = & 0^{2} + y^{2} \\ y^{4} & = & y^{2} \\ y^{4} - y^{2} & = & 0 \\ y^{2} (y^{2} - 1) & = & 0 \\ y^{2} (y - 1) (y + 1) & = & 0 \\ y & = & 0, 1, - 1 \end{array}$

3Step 3: Determine the symmetry.

Perform the test for symmetry with respect to the x-axis.

Replace $y$ with $- y$ .

$\begin{array}{rcl} {(x^{2} + (- y)^{2} - x)}^{2} & = & x^{2} + (- y)^{2} \\ {(x^{2} + y^{2} - x)}^{2} & = & x^{2} + y^{2} \end{array}$

The above equation is same as the given equation.

Perform the test for symmetry with respect to the $y$ -axis.

Replace $x$ with $- x$ .

$\begin{array}{rcl} {({(- x)}^{2} + y^{2} - (- x))}^{2} & = & {(- x)}^{2} + y^{2} \\ {(x^{2} + y^{2} + x)}^{2} & = & x^{2} + y^{2} \end{array}$

The above equation is not same as the given equation.

Perform the test for symmetry with respect to the origin.

Replace $x$ with $- x$ and $y$ with $- y$ .

$\begin{array}{rcl} {({(- x)}^{2} + (- y)^{2} - (- x))}^{2} & = & {(- x)}^{2} + (- y)^{2} \\ {(x^{2} + y^{2} + x)}^{2} & = & x^{2} + y^{2} \end{array}$