Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
