To find the equation of the circle with endpoints of a diameter at [tex]\( M(-1, 1) \)[/tex] and [tex]\( N(5, 9) \)[/tex], follow these steps:
1. Find the center of the circle:
The center of the circle is the midpoint of [tex]\( M \)[/tex] and [tex]\( N \)[/tex].
[tex]\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Here, [tex]\( M(-1, 1) \)[/tex] and [tex]\( N(5, 9) \)[/tex]:
[tex]\[
\text{Center} = \left( \frac{-1 + 5}{2}, \frac{1 + 9}{2} \right) = \left( \frac{4}{2}, \frac{10}{2} \right) = (2, 5)
\][/tex]
2. Calculate the radius of the circle:
The radius is half of the distance between [tex]\( M \)[/tex] and [tex]\( N \)[/tex]. First, find the distance between [tex]\( M \)[/tex] and [tex]\( N \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{(5 - (-1))^2 + (9 - 1)^2} = \sqrt{(5 + 1)^2 + (8)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\][/tex]
The radius is half of this distance:
[tex]\[
\text{Radius} = \frac{10}{2} = 5
\][/tex]
3. Write the equation of the circle:
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting the values we found:
[tex]\[
(x - 2)^2 + (y - 5)^2 = 5^2
\][/tex]
[tex]\[
(x - 2)^2 + (y - 5)^2 = 25
\][/tex]
From these steps, we see that the correct equation for the circle is:
[tex]\((x-2)^2+(y-5)^2=25\)[/tex].
Thus, the correct option for this problem is:
[tex]\((x-2)^2+(y-5)^2=25\)[/tex].
