Which Equation Represents A Circle That Contains The Point { (-5, -3)$}$ And Has A Center At { (-2, 1)$}$?A. { (x-1)^2 + (y+2)^2 = 25$}$B. { (x+2)^2 + (y-1)^2 = 5$}$C. { (x+2)^2 + (y-1)^2 = 25$}$D.
Introduction
In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle is a mathematical representation of this concept, and it is used to describe the shape and size of the circle. In this article, we will explore the equation of a circle that contains a specific point and has a center at a given location.
Understanding the Equation of a Circle
The equation of a circle is given by the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle, and r is the radius of the circle. This equation represents a circle that is centered at (h, k) and has a radius of r.
Given Information
We are given that the circle contains the point (-5, -3) and has a center at (-2, 1). We need to find the equation of the circle that satisfies these conditions.
Step 1: Identify the Center of the Circle
The center of the circle is given as (-2, 1). This means that the value of h is -2, and the value of k is 1.
Step 2: Identify the Radius of the Circle
To find the radius of the circle, we need to calculate the distance between the center of the circle and the given point. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center of the circle, and (x2, y2) is the given point.
Plugging in the values, we get:
d = √((-5 - (-2))^2 + (-3 - 1)^2) d = √((-3)^2 + (-4)^2) d = √(9 + 16) d = √25 d = 5
Therefore, the radius of the circle is 5.
Step 3: Write the Equation of the Circle
Now that we have the center and radius of the circle, we can write the equation of the circle using the formula:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values, we get:
(x - (-2))^2 + (y - 1)^2 = 5^2 (x + 2)^2 + (y - 1)^2 = 25
Conclusion
In this article, we have explored the equation of a circle that contains a specific point and has a center at a given location. We have identified the center and radius of the circle and used this information to write the equation of the circle. The equation of the circle is given by:
(x + 2)^2 + (y - 1)^2 = 25
This equation represents a circle that contains the point (-5, -3) and has a center at (-2, 1).
Answer
The correct answer is:
Introduction
In our previous article, we explored the equation of a circle that contains a specific point and has a center at a given location. We identified the center and radius of the circle and used this information to write the equation of the circle. In this article, we will answer some frequently asked questions about the equation of a circle.
Q: What is the general form of the equation of a circle?
A: The general form of the equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle, and r is the radius of the circle.
Q: How do I identify the center of the circle from the equation?
A: To identify the center of the circle from the equation, you need to look for the values of h and k. The center of the circle is given by the coordinates (h, k).
Q: How do I identify the radius of the circle from the equation?
A: To identify the radius of the circle from the equation, you need to look for the value of r^2. The radius of the circle is given by the square root of r^2.
Q: What is the difference between the equation of a circle and the equation of an ellipse?
A: The equation of a circle is given by:
(x - h)^2 + (y - k)^2 = r^2
while the equation of an ellipse is given by:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where a and b are the semi-major and semi-minor axes of the ellipse.
Q: Can I write the equation of a circle in different forms?
A: Yes, you can write the equation of a circle in different forms. For example, you can write the equation of a circle in the form:
(x - h)^2 + (y - k)^2 = r^2
or
(x - h)^2 - (y - k)^2 = r^2
or
(x - h)^2 + (y - k)^2 - r^2 = 0
Q: How do I graph a circle using its equation?
A: To graph a circle using its equation, you need to plot the center of the circle and then draw a circle with a radius equal to the value of r.
Q: Can I use the equation of a circle to find the distance between two points?
A: Yes, you can use the equation of a circle to find the distance between two points. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Conclusion
In this article, we have answered some frequently asked questions about the equation of a circle. We have discussed the general form of the equation of a circle, how to identify the center and radius of the circle, and how to graph a circle using its equation. We have also discussed how to use the equation of a circle to find the distance between two points.
Additional Resources
Answer Key
- Q1: (x - h)^2 + (y - k)^2 = r^2
- Q2: (h, k)
- Q3: √r^2
- Q4: No
- Q5: Yes
- Q6: Yes