Question

1. write an equation for each circle.

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.

Step2: Identify the center of the circles

From the graph, the center of both circles is $(0,0)$ (since the circles are centered at the origin of the coordinate - plane).

Step3: Find the radius of the inner circle

The inner circle passes through the point $(0,2)$ and $(2,0)$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ from the center $(0,0)$ to a point on the circle (e.g., $(2,0)$), we have $r_1=\sqrt{(2 - 0)^2+(0 - 0)^2}=2$. So the equation of the inner circle is $x^{2}+y^{2}=4$ (since $h = 0,k = 0,r = 2$ and $(x - 0)^2+(y - 0)^2=2^2$).

Step4: Find the radius of the outer circle

The outer circle passes through the point $(0,4)$ and $(4,0)$. Using the distance formula from the center $(0,0)$ to a point on the circle (e.g., $(4,0)$), we have $r_2=\sqrt{(4 - 0)^2+(0 - 0)^2}=4$. So the equation of the outer circle is $x^{2}+y^{2}=16$ (since $h = 0,k = 0,r = 4$ and $(x - 0)^2+(y - 0)^2=4^2$).

Answer:

The equation of the inner circle is $x^{2}+y^{2}=4$, and the equation of the outer circle is $x^{2}+y^{2}=16$.