The Solution:
Given the ends of a diameter:
[tex](17,12)\text{ and }(13,16)[/tex]Required:
To write the equation of the circle.
Step 1:
Find the center of the circle by using the midpoint formula.
[tex]Midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]In this case,
[tex]\begin{gathered} x_1=17 \\ y_1=12 \\ x_2=13 \\ y_2=16 \end{gathered}[/tex]Substituting, we get
[tex]\begin{gathered} \text{ Midpoint}=(\frac{17+13}{2},\frac{12+16}{2})=(\frac{30}{2},\frac{28}{2})=(15,14) \\ So, \\ The\text{ center}=(15,14) \end{gathered}[/tex]Step 2:
Find the radius of the circle.
Using the distance between two points formula:
[tex]r^2=(x_2-x_1)^2+(y_2-y_1)^2[/tex][tex]\begin{gathered} r^2=(13-17)^2+(16-12)^2 \\ \\ r^2=(-4)^2+(4)^2 \\ \\ r^2=16+16 \\ \\ r^2=32 \end{gathered}[/tex]Step 3:
Write the equation of the circle.
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ \text{ Where} \\ h=15 \\ k=14 \\ r=32 \end{gathered}[/tex][tex]\begin{gathered} (x-15)^2+(y-14)^2=32^2 \\ \\ (x-15)^2+(y-14)^2-1024=0 \end{gathered}[/tex]Therefore, the correct answers are:
[tex]\begin{gathered} center=(15,14) \\ \\ Equation:\text{ }(x-15)^2+(y-14)^2-1024=0 \end{gathered}[/tex]