Question

circle a and b are graphed in the xy - plane. circle a is represented by the equation(x - 9)^2+(y + 9)^2 = 81 and intersects the x - axis at the point (r, s). circle b has its center at (r, s) and has a radius of the same length as circle a. which equation represents circle b? a. x^2+(y - 9)^2 = 81 b. x^2+(y + 9)^2 = 81 c. (x - 9)^2+y^2 = 81 d. (x + 9)^2+y^2 = 81

Explanation:

Step1: Recall the standard - form of a circle equation

The standard - form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. For circle A with the equation $(x - 9)^2+(y + 9)^2 = 81$, we can identify that the center of circle A is $(9,-9)$ and the radius $r=\sqrt{81}=9$.

Step2: Analyze each option

• Option a: The equation $x^{2}+(y - 9)^{2}=81$ has a center at $(0,9)$ and radius $r = 9$.
• Option b: The equation $x^{2}+(y + 9)^{2}=81$ has a center at $(0,-9)$ and radius $r = 9$.
• Option c: The equation $(x - 9)^{2}+y^{2}=81$ has a center at $(9,0)$ and radius $r = 9$.
• Option d: The equation $(x + 9)^{2}+y^{2}=81$ has a center at $(-9,0)$ and radius $r = 9$. Since circle B has its center at $(r,s)$ and intersects the x - axis at $(r,s)$ and has the same radius as circle A ($r = 9$), and circle A has center $(9,-9)$ and radius 9. If circle B has center on the x - axis and radius 9, the equation of circle B can be $(x - 9)^{2}+y^{2}=81$ (assuming the center of circle B is on the positive x - axis).

Answer:

C. $(x - 9)^{2}+y^{2}=81$