Which Equation Represents A Circle With A Center At { (-3, -5)$}$ And A Radius Of 6 Units?A. { (x - 3)^2 + (y - 5)^2 = 6$}$ B. { (x - 3)^2 + (y - 5)^2 = 36$}$ C. { (x + 3)^2 + (y + 5)^2 = 6$}$ D. [$(x +

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point, known as the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore the equation of a circle and how to represent it mathematically.

The General Equation of a Circle

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

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

where ${(x, y)}$ represents any point on the circle.

Example: Circle with Center at ${(-3, -5)}$ and Radius of 6 Units

We are given that the center of the circle is at ${(-3, -5)}$ and the radius is 6 units. We need to find the equation of this circle.

Step 1: Identify the Center and Radius

The center of the circle is given as ${(-3, -5)}$, which means ${h = -3}$ and ${k = -5}$. The radius is given as 6 units, which means ${r = 6}$.

Step 2: Plug in the Values into the General Equation

Substituting the values of ${h}$, ${k}$, and ${r}$ into the general equation, we get:

${(x - (-3))^2 + (y - (-5))^2 = 6^2}$

Simplifying the equation, we get:

${(x + 3)^2 + (y + 5)^2 = 36}$

Conclusion

Therefore, the equation that represents a circle with a center at ${(-3, -5)}$ and a radius of 6 units is:

${(x + 3)^2 + (y + 5)^2 = 36}$

This equation satisfies the conditions of the problem, and it represents a circle with the given center and radius.

Comparison with the Given Options

Let's compare our derived equation with the given options:

A. ${(x - 3)^2 + (y - 5)^2 = 6}$ B. ${(x - 3)^2 + (y - 5)^2 = 36}$ C. ${(x + 3)^2 + (y + 5)^2 = 6}$ D. ${(x + 3)^2 + (y + 5)^2 = 36}$

Our derived equation matches option D.

Final Answer

The final answer is:

${(x + 3)^2 + (y + 5)^2 = 36}$

$$

Introduction

In our previous article, we explored the equation of a circle and how to represent it mathematically. We also derived the equation of a circle with a center at ${(-3, -5)}$ and a radius of 6 units. In this article, we will answer some frequently asked questions about circle equations.

Q: What is the general equation of a circle?

A: The general 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 identify the center and radius of a circle from its equation?

A: To identify the center and radius of a circle from its equation, you need to look for the values of ${h}$, ${k}$, and ${r}$ in the equation. The values of ${h}$ and ${k}$ represent the coordinates of the center, and the value of ${r}$ represents the radius.

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

A: The radius in a circle equation represents the distance from the center to any point on the circle. It is a measure of the size of the circle.

Q: How do I determine if a point lies on a circle?

A: To determine if a point lies on a circle, you need to substitute the coordinates of the point into the equation of the circle and check if the equation is satisfied. If the equation is satisfied, then the point lies on the circle.

Q: Can a circle have a negative radius?

A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value, representing the distance from the center to any point on the circle.

Q: What is the equation of a circle with a center at the origin and a radius of 4 units?

A: The equation of a circle with a center at the origin and a radius of 4 units is given by:

${x^2 + y^2 = 16}$

Q: How do I graph a circle from its equation?

A: To graph a circle from its equation, you need to plot the center of the circle and then draw a circle with the given radius. You can use a compass or a graphing calculator to help you draw the circle.

Q: Can a circle have a center at a point with negative coordinates?

A: Yes, a circle can have a center at a point with negative coordinates. For example, the equation of a circle with a center at ${(-3, -5)}$ and a radius of 6 units is given by:

${(x + 3)^2 + (y + 5)^2 = 36}$

Conclusion

In this article, we answered some frequently asked questions about circle equations. We hope that this guide has been helpful in understanding the concept of circle equations and how to apply them in different situations.

Common Mistakes to Avoid

• Not identifying the center and radius of a circle from its equation.
• Not checking if a lies on a circle by substituting its coordinates into the equation.
• Assuming that a circle can have a negative radius.
• Not graphing a circle from its equation correctly.

Final Tips

• Practice deriving the equation of a circle from its center and radius.
• Practice graphing a circle from its equation.
• Use a compass or a graphing calculator to help you draw a circle.
• Check if a point lies on a circle by substituting its coordinates into the equation.