Select The Correct Answer.The Points J ( − 8 , 9 J(-8,9 J ( − 8 , 9 ] And K ( − 2 , − 5 K(-2,-5 K ( − 2 , − 5 ] Are Endpoints Of A Diameter Of Circle C. Which Equation Represents Circle C?A. ( X − 5 ) 2 + ( Y + 2 ) 2 = 58 (x-5)^2+(y+2)^2=58 ( X − 5 ) 2 + ( Y + 2 ) 2 = 58 B. ( X − 5 ) 2 + ( Y + 2 ) 2 = 232 (x-5)^2+(y+2)^2=232 ( X − 5 ) 2 + ( Y + 2 ) 2 = 232 C.

Introduction

In this problem, we are given the endpoints of a diameter of a circle, and we need to find the equation that represents the circle. The points $J(-8,9)$ and $K(-2,-5)$ are the endpoints of the diameter. To find the equation of the circle, we need to use the center and radius of the circle.

Step 1: Find the Center of the Circle

The center of the circle is the midpoint of the diameter. To find the midpoint, we need to average the x-coordinates and the y-coordinates of the endpoints.

# Define the coordinates of the endpoints
x1 = -8
y1 = 9
x2 = -2
y2 = -5
center_x = (x1 + x2) / 2

center_y = (y1 + y2) / 2

Step 2: Find the Radius of the Circle

The radius of the circle is half the length of the diameter. To find the length of the diameter, we need to use the distance formula.

# Import the math module
import math

diameter_length = math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

radius = diameter_length / 2

Step 3: Write the Equation of the Circle

The equation of a circle with center $(h,k)$ and radius $r$ is given by:

$(x-h)^2 + (y-k)^2 = r^2$

We can now substitute the values of the center and radius into this equation.

# Define the equation of the circle
equation = f"(x-{center_x})^2 + (y-{center_y})^2 = {radius**2}"

Step 4: Simplify the Equation

We can simplify the equation by evaluating the expressions.

# Simplify the equation
simplified_equation = f"(x+5)^2 + (y+7)^2 = 58"

Conclusion

The equation that represents circle C is:

$(x+5)^2 + (y+7)^2 = 58$

This equation is in the standard form of a circle, which is $(x-h)^2 + (y-k)^2 = r^2$. The center of the circle is $(-5,-7)$, and the radius is $\sqrt{58}$.

Answer

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Introduction

In our previous article, we explored the equation of a circle with endpoints of a diameter given. In this article, we will delve deeper into the world of circles and answer some frequently asked questions.

Q&A: Circle C

Q: What is the center of the circle?

A: The center of the circle is the midpoint of the diameter. To find the midpoint, we need to average the x-coordinates and the y-coordinates of the endpoints.

Q: How do I find the radius of the circle?

A: The radius of the circle is half the length of the diameter. To find the length of the diameter, we need to use the distance formula.

Q: What is the equation of a circle?

A: The equation of a circle with center $(h,k)$ and radius $r$ is given by:

$(x-h)^2 + (y-k)^2 = r^2$

Q: How do I simplify the equation of a circle?

A: We can simplify the equation by evaluating the expressions.

Q: What is the standard form of a circle?

A: The standard form of a circle is $(x-h)^2 + (y-k)^2 = r^2$. The center of the circle is $(h,k)$, and the radius is $r$.

Q: How do I find the length of the diameter?

A: To find the length of the diameter, we need to use the distance formula.

Q: What is the distance formula?

A: The distance formula is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Q: How do I find the midpoint of a line segment?

A: To find the midpoint of a line segment, we need to average the x-coordinates and the y-coordinates of the endpoints.

Q: What is the midpoint formula?

A: The midpoint formula is given by:

$(x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Q: How do I find the equation of a circle with a given center and radius?

A: To find the equation of a circle with a given center and radius, we need to substitute the values of the center and radius into the equation of a circle.

Q: What is the equation of a circle with center $(h,k)$ and radius $r$?

A: The equation of a circle with center $(h,k)$ and radius $r$ is given by:

$(x-h)^2 + (y-k)^2 = r^2$

Q: How do I find the radius of a circle?

A: To find the radius of a circle, we need to use the equation of a circle and solve for the radius.

Q: What is the radius of a circle with equation $(x-5)^2 + (y+7)^2 = 58$?

A: The radius of a circle with equation $(x-5)^2 + (y+7)^2 = 58$ is $\sqrt{58}$.

Conclusion

In this article, we have answered some frequently asked questions about circles. We have explored the equation of a circle, the center and radius of a circle, and the distance formula. We have also provided examples and formulas to help you understand the concepts.

Additional Resources

• Circle Equation Calculator
• Circle Formula
• Distance Formula

Final Thoughts

Circles are an essential concept in mathematics, and understanding the equation of a circle is crucial for solving problems in geometry and trigonometry. We hope that this article has provided you with a comprehensive guide to circles and has helped you to understand the concepts better.