Select The Correct Answer.Which Equation Represents A Circle With A Center At $(-5,5)$ And A Radius Of 3 Units?A. $(x-5)^2+(y+5)^2=3$B. $(x+5)^2+(y-5)^2=6$C. \$(x+5)^2+(y-5)^2=3$[/tex\]D.

When it comes to representing a circle on a coordinate plane, the equation of a circle is a crucial concept to grasp. The equation of a circle with a center at $(h, k)$ and a radius of $r$ units is given by $(x - h)^2 + (y - k)^2 = r^2$. In this article, we will explore how to select the correct equation that represents a circle with a center at $(-5, 5)$ and a radius of 3 units.

The General Equation of a Circle

The general equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the center of the circle and $r$ represents the radius. To find the equation of a circle, we need to substitute the values of the center and the radius into the general equation.

Substituting the Values

Let's substitute the values of the center and the radius into the general equation. The center of the circle is $(-5, 5)$, and the radius is 3 units. We can substitute these values into the general equation as follows:

$(x - (-5))^2 + (y - 5)^2 = 3^2$

Simplifying the equation, we get:

$(x + 5)^2 + (y - 5)^2 = 9$

Comparing the Options

Now that we have the correct equation of the circle, let's compare it with the options provided:

A. $(x-5)^2+(y+5)^2=3$ B. $(x+5)^2+(y-5)^2=6$ C. $(x+5)^2+(y-5)^2=3$ D. $(x-5)^2+(y-5)^2=3$

From the options provided, only one equation matches the correct equation of the circle that we derived earlier.

The Correct Answer

The correct answer is:

C. $(x+5)^2+(y-5)^2=3$

This equation represents a circle with a center at $(-5, 5)$ and a radius of 3 units.

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics that is used to represent a circle on a coordinate plane. By substituting the values of the center and the radius into the general equation, we can derive the correct equation of the circle. In this article, we explored how to select the correct equation that represents a circle with a center at $(-5, 5)$ and a radius of 3 units. We compared the options provided and found that only one equation matches the correct equation of the circle.

Frequently Asked Questions

Q: What is the general equation of a circle?

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

Q: How do I substitute the values of the center and the radius into the general equation?

A: To substitute the values of the center and the radius into the general equation, simply replace $h$ with the x-coordinate of the center, $k$ with the y-coordinate of the center, and $r$ with the radius.

Q: What is the correct equation of a circle with a center at $(-5, 5)$ and a radius of 3 units?

A: The correct equation of a circle with a center at $(-5, 5)$ and a radius of 3 units is $(x + 5)^2 + (y - 5)^2 = 9$.

Q: How do I compare the options provided to find the correct equation of the circle?

A: To compare the options provided, simply substitute the values of the center and the radius into each equation and check if it matches the correct equation of the circle.

Additional Resources

For more information on the equation of a circle, you can refer to the following resources:

• Khan Academy: Equation of a Circle
• Mathway: Equation of a Circle
• Wolfram Alpha: Equation of a Circle

Conclusion

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In our previous article, we explored how to select the correct equation that represents a circle with a center at $(-5, 5)$ and a radius of 3 units. We also discussed the general equation of a circle and how to substitute the values of the center and the radius into the equation. In this article, we will answer some frequently asked questions about the equation of a circle.

Q&A

Q: What is the general equation of a circle?

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

Q: How do I substitute the values of the center and the radius into the general equation?

A: To substitute the values of the center and the radius into the general equation, simply replace $h$ with the x-coordinate of the center, $k$ with the y-coordinate of the center, and $r$ with the radius.

Q: What is the correct equation of a circle with a center at $(-5, 5)$ and a radius of 3 units?

A: The correct equation of a circle with a center at $(-5, 5)$ and a radius of 3 units is $(x + 5)^2 + (y - 5)^2 = 9$.

Q: How do I compare the options provided to find the correct equation of the circle?

A: To compare the options provided, simply substitute the values of the center and the radius into each equation and check if it matches the correct equation of the circle.

Q: What is the significance of the center and the radius in the equation of a circle?

A: The center and the radius are the two most important components of the equation of a circle. The center represents the point around which the circle is centered, while the radius represents the distance from the center to the edge of the circle.

Q: Can I use the equation of a circle to find the area of a circle?

A: Yes, you can use the equation of a circle to find the area of a circle. The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle.

Q: How do I find the equation of a circle with a center at $(2, 3)$ and a radius of 4 units?

A: To find the equation of a circle with a center at $(2, 3)$ and a radius of 4 units, simply substitute the values of the center and the radius into the general equation:

$(x - 2)^2 + (y - 3)^2 = 4^2$

Simplifying the equation, we get:

$(x - 2)^2 + (y - 3)^2 = 16$

Q: Can I use the equation of a circle to find the circumference of a circle?

A: Yes, you can use the equation of a circle to find the circumference of a circle. The circumference of a circle is given by the formula $C = 2\pi r$, where $r$ is the radius of the circle.

Q: How do I find the equation of a circle with a center at $(-1, 2)$ and a radius of 5 units?

A: To find the equation of a circle with a center at $(-1, 2)$ and a radius of 5 units, simply substitute the values of the center and the radius into the general equation:

$(x + 1)^2 + (y - 2)^2 = 5^2$

Simplifying the equation, we get:

$(x + 1)^2 + (y - 2)^2 = 25$

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics that is used to represent a circle on a coordinate plane. By substituting the values of the center and the radius into the general equation, we can derive the correct equation of the circle. In this article, we answered some frequently asked questions about the equation of a circle, including how to substitute the values of the center and the radius into the general equation, how to compare the options provided to find the correct equation of the circle, and how to find the area and circumference of a circle.

Additional Resources

For more information on the equation of a circle, you can refer to the following resources:

• Khan Academy: Equation of a Circle
• Mathway: Equation of a Circle
• Wolfram Alpha: Equation of a Circle

Practice Problems

1. Find the equation of a circle with a center at $(3, 4)$ and a radius of 2 units.
2. Find the equation of a circle with a center at $(-2, 1)$ and a radius of 3 units.
3. Find the area of a circle with a radius of 5 units.
4. Find the circumference of a circle with a radius of 4 units.

Answer Key

1. $(x - 3)^2 + (y - 4)^2 = 2^2$
2. $(x + 2)^2 + (y - 1)^2 = 3^2$
3. $A = \pi (5)^2 = 25\pi$
4. $C = 2\pi (4) = 8\pi$