To determine the equation of the circle with a given diameter having endpoints [tex]\((-3, 7)\)[/tex] and [tex]\((5, 7)\)[/tex], we need to follow a few steps. Here's a detailed, step-by-step solution.
### Step 1: Find the Center of the Circle
The center of the circle is the midpoint of the diameter. We can find this by calculating the midpoint of the given endpoints [tex]\((-3, 7)\)[/tex] and [tex]\((5, 7)\)[/tex].
The midpoint formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Plugging in our points:
[tex]\[ x_{\text{center}} = \frac{-3 + 5}{2} = \frac{2}{2} = 1 \][/tex]
[tex]\[ y_{\text{center}} = \frac{7 + 7}{2} = \frac{14}{2} = 7 \][/tex]
So, the center of the circle is [tex]\((1, 7)\)[/tex].
### Step 2: Calculate the Radius of the Circle
The radius is half the length of the diameter. To find the length of the diameter, we use the distance formula between the points [tex]\((-3, 7)\)[/tex] and [tex]\((5, 7)\)[/tex].
The distance formula is:
[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Plugging in our points:
[tex]\[ \text{diameter} = \sqrt{(5 - (-3))^2 + (7 - 7)^2} \][/tex]
[tex]\[ = \sqrt{(5 + 3)^2 + 0^2} \][/tex]
[tex]\[ = \sqrt{8^2 + 0} \][/tex]
[tex]\[ = \sqrt{64} \][/tex]
[tex]\[ = 8 \][/tex]
Since the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{diameter}{2} = \frac{8}{2} = 4 \][/tex]
### Step 3: Write the Equation of the Circle
The general equation for a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Plugging in our center [tex]\((1, 7)\)[/tex] and radius [tex]\(4\)[/tex]:
[tex]\[ (x - 1)^2 + (y - 7)^2 = 4^2 \][/tex]
[tex]\[ (x - 1)^2 + (y - 7)^2 = 16 \][/tex]
Thus, the equation of the circle is:
[tex]\[ (x - 1)^2 + (y - 7)^2 = 16 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{(x-1)^2+(y-7)^2=16} \][/tex]
