Answer:
[tex]{(x - 1)^2 + (y - 6)^2 = 17}[/tex]
Step-by-step explanation:
We must start by finding the center of the circle using the midpoint formula. From there we can use the distance formula to find the radius. Having the center and the radius will be used to get the equation of the circle.
[tex]\fbox{ \parbox{\textwidth}{ \vspace{0.5em} % Space to lower the text \textbf{Midpoint Formula:} The midpoint \( M \) of points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] }}[/tex]
[tex]\fbox{ \parbox{\textwidth}{ \vspace{0.5em} % Space to lower the text \textbf{Distance Formula:} The distance \( d \) between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] }}[/tex]
[tex]\fbox{ \parbox{\textwidth}{ \vspace{0.5em} % Space to lower the text \textbf{General Form of a Circle:} The equation of a circle with center \( (h, k) \) and radius \( r \) is: \[ (x - h)^2 + (y - k)^2 = r^2 \] }}[/tex]
Solving:
[tex]\subsection*{Center of the Circle:}Plug in \((-3, 7)\) and \((5, 5)\) into the midpoint formula:\[h = \frac{-3 + 5}{2} = \frac{2}{2} = 1\]\[k = \frac{7 + 5}{2} = \frac{12}{2} = 6\]\\\\The center of the circle is at the point (1,6)[/tex]
[tex]\subsection*{Radius of the Circle:} Using the center \((1, 6)\) and the endpoint \((-3, 7)\), we apply the distance formula:\[r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]\[r = \sqrt{(-3 - 1)^2 + (7 - 6)^2}\]\[r = \sqrt{(-4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17}\][/tex]
[tex]\subsection*{Equation of the Circle:}Plug the values \(h = 1\), \(k = 6\), and \(r^2 = 17\) into the general equation of the circle:\[\boxed{(x - 1)^2 + (y - 6)^2 = 17}\][/tex]
Therefore, the equation of the circle is : [tex]{(x - 1)^2 + (y - 6)^2 = 17}[/tex]