Explanation
The equation of a circle with center (h,k) and radius r units is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]then
Step 1
find the diameter of the cirlce:
to do this, we can use the distance between two points formula:
if
[tex]\begin{gathered} A\mleft(x_1,y_1\mright) \\ B(x_2,y_2) \end{gathered}[/tex]the distance from A to B is
[tex]\begin{gathered} d_{AB}=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \end{gathered}[/tex]so,let
distance CD=
[tex]\begin{gathered} CD=\sqrt[]{(1-(-7))^2+(2-(-4))^2} \\ CD=\sqrt[]{(8)^2+(6)^2} \\ CD=\sqrt[]{64+36} \\ CD=\sqrt[]{100} \\ CD=10 \end{gathered}[/tex]hence, the diameter of the circle is
[tex]\begin{gathered} \text{diameter}=10 \\ w\text{e know also} \\ \text{diameter}=2\cdot\text{raidus} \\ \frac{\text{diamteter}}{2}=radius \\ \text{replace} \\ \frac{10}{2}=\text{ radius} \\ \text{radius}=5 \end{gathered}[/tex]Step 2
find the center of the circle:
the center of the circle is the midpoint of CD
so
[tex]\text{midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]replace
[tex]\begin{gathered} \text{midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ \text{midpoint}=(\frac{-7+1}{2},\frac{-4+2_{}}{2}) \\ \text{midpoint}=(\frac{-6}{2},\frac{-2}{2}) \\ \text{midpoint}=(-3,-1) \\ \end{gathered}[/tex]so, the center of the circle is (-3,-1)
Step 3
finally, replace in the formula to get the equation of the circle
let
[tex]\begin{gathered} center=\mleft(-3,-1\mright) \\ radius=5 \end{gathered}[/tex]replace
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ (x-(-3))^2+(y-(-1))^2=5^2 \\ (x+3)^2+(y+1)^2=25 \end{gathered}[/tex]therefore, the answer is
[tex](x+3)^2+(y+1)^2=25[/tex]I hope this helps you
