Question

in studios and 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 $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$. (a) find the intercepts of the graph of the equation. (b) test for symmetry with respect to the x - axis, y - axis, and origin. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the intercept(s) is/are (type an ordered pair. use a comma to separate answers as needed. type each answer only once.) b. there are no intercepts.

Explanation:

Step1: Find x - intercepts

Set $y = 0$ in the equation $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$.
When $y = 0$, the equation becomes $(x^{2}+0 - 0)^{2}=4x^{2}+0$, i.e., $x^{4}-4x^{2}=0$. Let $u = x^{2}$, then $u^{2}-4u=0$. Factoring gives $u(u - 4)=0$. So $u=0$ or $u = 4$. When $u = 0$, $x = 0$; when $u = 4$, $x=\pm2$. The x - intercepts are $(0,0),(2,0),(-2,0)$.

Step2: Find y - intercepts

Set $x = 0$ in the equation $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$.
When $x = 0$, the equation becomes $(y^{2}-2y)^{2}=4y^{2}$. Expanding, $y^{4}-4y^{3}+4y^{2}-4y^{2}=0$, so $y^{4}-4y^{3}=0$. Factoring out $y^{3}$ gives $y^{3}(y - 4)=0$. So $y = 0$ or $y = 4$. The y - intercepts are $(0,0),(0,4)$.

Step3: Test for x - axis symmetry

Replace $y$ with $-y$ in the equation $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$.
We get $(x^{2}+y^{2}+2y)^{2}=4x^{2}+4y^{2}$, which is not the same as the original equation, so there is no x - axis symmetry.

Step4: Test for y - axis symmetry

Replace $x$ with $-x$ in the equation $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$.
We get $((-x)^{2}+y^{2}-2y)^{2}=4(-x)^{2}+4y^{2}$, which simplifies to $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$, the same as the original equation, so there is y - axis symmetry.

Step5: Test for origin symmetry

Replace $x$ with $-x$ and $y$ with $-y$ in the equation $(x^{2}+y^{2}-2y)^{2}=4x^{2}+4y^{2}$.
We get $(x^{2}+y^{2}+2y)^{2}=4x^{2}+4y^{2}$, which is not the same as the original equation, so there is no origin symmetry.

Answer:

(a) The intercepts are $(0,0),(2,0),(-2,0),(0,4)$.
(b) No x - axis symmetry; y - axis symmetry; no origin symmetry.