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To find the equation of the circle, we need to determine the center and the radius of the circle. Given that the points [tex]\( \wedge(-8, 9) \)[/tex] and [tex]\( K(-2, -5) \)[/tex] are the endpoints of the diameter, we can follow these steps to find the necessary components of the circle's equation.
1. Find the Center (Midpoint) of the Circle:
The center of the circle is the midpoint of the given diameter endpoints. 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]
Substituting the provided points, [tex]\( \wedge(-8, 9) \)[/tex] and [tex]\( K(-2, -5) \)[/tex]:
[tex]\[ \left( \frac{-8 + (-2)}{2}, \frac{9 + (-5)}{2} \right) = \left( \frac{-10}{2}, \frac{4}{2} \right) = (-5, 2) \][/tex]
So, the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-5, 2)\)[/tex].
2. Calculate the Radius:
The radius is half the length of the diameter. To find the length of the diameter, we use the distance formula between the endpoints [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex]:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the provided points:
[tex]\[ d = \sqrt{(-2 - (-8))^2 + (-5 - 9)^2} = \sqrt{(-2 + 8)^2 + (-5 - 9)^2} = \sqrt{6^2 + (-14)^2} = \sqrt{36 + 196} = \sqrt{232} \][/tex]
The diameter is [tex]\( \sqrt{232} \)[/tex], so the radius [tex]\( r \)[/tex] is half of this:
[tex]\[ r = \frac{\sqrt{232}}{2} \][/tex]
Squaring the radius to use in the circle's equation:
[tex]\[ r^2 = \left( \frac{\sqrt{232}}{2} \right)^2 = \frac{232}{4} = 58 \][/tex]
3. Write the Equation of the Circle:
The standard equation of a circle with center [tex]\((h,k)\)[/tex] and radius squared [tex]\(r^2\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Substituting [tex]\(h = -5\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(r^2 = 58\)[/tex]:
[tex]\[ (x - (-5))^2 + (y - 2)^2 = 58 \][/tex]
Simplifying:
[tex]\[ (x + 5)^2 + (y - 2)^2 = 58 \][/tex]
So, the equation of the circle is [tex]\((x + 5)^2 + (y - 2)^2 = 58\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C. \ (x+5)^2 + (y-2)^2 = 58} \][/tex]
1. Find the Center (Midpoint) of the Circle:
The center of the circle is the midpoint of the given diameter endpoints. 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]
Substituting the provided points, [tex]\( \wedge(-8, 9) \)[/tex] and [tex]\( K(-2, -5) \)[/tex]:
[tex]\[ \left( \frac{-8 + (-2)}{2}, \frac{9 + (-5)}{2} \right) = \left( \frac{-10}{2}, \frac{4}{2} \right) = (-5, 2) \][/tex]
So, the center of the circle [tex]\((h, k)\)[/tex] is [tex]\((-5, 2)\)[/tex].
2. Calculate the Radius:
The radius is half the length of the diameter. To find the length of the diameter, we use the distance formula between the endpoints [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex]:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the provided points:
[tex]\[ d = \sqrt{(-2 - (-8))^2 + (-5 - 9)^2} = \sqrt{(-2 + 8)^2 + (-5 - 9)^2} = \sqrt{6^2 + (-14)^2} = \sqrt{36 + 196} = \sqrt{232} \][/tex]
The diameter is [tex]\( \sqrt{232} \)[/tex], so the radius [tex]\( r \)[/tex] is half of this:
[tex]\[ r = \frac{\sqrt{232}}{2} \][/tex]
Squaring the radius to use in the circle's equation:
[tex]\[ r^2 = \left( \frac{\sqrt{232}}{2} \right)^2 = \frac{232}{4} = 58 \][/tex]
3. Write the Equation of the Circle:
The standard equation of a circle with center [tex]\((h,k)\)[/tex] and radius squared [tex]\(r^2\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Substituting [tex]\(h = -5\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(r^2 = 58\)[/tex]:
[tex]\[ (x - (-5))^2 + (y - 2)^2 = 58 \][/tex]
Simplifying:
[tex]\[ (x + 5)^2 + (y - 2)^2 = 58 \][/tex]
So, the equation of the circle is [tex]\((x + 5)^2 + (y - 2)^2 = 58\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C. \ (x+5)^2 + (y-2)^2 = 58} \][/tex]
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