Question

determining equations of concentric circles
concentric circles are circles with the same center but different radii. which equations represent concentric circles along with the circle shown in the graph? check all that apply.
□ $x^{2}+y^{2}=25$
□ $(x - 8)^{2}+(y - 9)^{2}=3$
□ $(x - 8)^{2}+(y - 9)^{2}=14$
□ $(x - 8)^{2}+(y + 9)^{2}=3$
□ $(x + 8)^{2}+(y + 9)^{2}=25$
□ $(x + 9)^{2}+(y + 8)^{2}=3$

Explanation:

Step1: Recall circle - equation formula

The standard form of the equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. Concentric circles have the same $(h,k)$.

Step2: Determine the center of the given circle

By observing the graph, assume the center of the given circle is $(8,9)$.

Step3: Check each option

• For $x^{2}+y^{2}=25$, the center is $(0,0)$, so it is not concentric.
• For $(x - 8)^2+(y - 9)^2=3$, the center is $(8,9)$, so it is concentric.
• For $(x - 8)^2+(y - 9)^2=14$, the center is $(8,9)$, so it is concentric.
• For $(x - 8)^2+(y + 9)^2=3$, the center is $(8,-9)$, so it is not concentric.
• For $(x + 8)^2+(y + 9)^2=25$, the center is $(-8,-9)$, so it is not concentric.
• For $(x + 9)^2+(y + 8)^2=3$, the center is $(-9,-8)$, so it is not concentric.

Answer:

$(x - 8)^2+(y - 9)^2=3$, $(x - 8)^2+(y - 9)^2=14$