Question

which of the following equations will produce the graph shown below? a. 4x^2 + 4y^2 = 64 b. 20x^2 - 20y^2 = 400

Explanation:

Step1: Recall circle - equation form

The standard form of the equation of a circle is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center of the circle and $r$ is the radius. For a circle centered at the origin $(0,0)$, the equation is $x^{2}+y^{2}=r^{2}$. We can also rewrite given equations in standard - like forms.

Step2: Analyze option A

For the equation $4x^{2}+4y^{2}=64$, divide both sides by 4. We get $x^{2}+y^{2}=16$, which is in the form of a circle equation $x^{2}+y^{2}=r^{2}$ with $r = 4$. The graph of $x^{2}+y^{2}=16$ is a circle centered at the origin with radius $r = 4$.

Step3: Analyze option B

For the equation $20x^{2}-20y^{2}=400$, divide both sides by 400. We get $\frac{x^{2}}{20}-\frac{y^{2}}{20}=1$, which is the equation of a hyperbola, not a circle.

Answer:

A. $4x^{2}+4y^{2}=64$