Type The Correct Answer In Each Box.A Circle Is Centered At The Point { (5, -4)$}$ And Passes Through The Point { (-3, 2)$}$.The Equation Of This Circle Is { (x + \square)^2 + (y + \square)^2 = \qquad$}$
Introduction
In mathematics, a circle is a set of points that are 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 position of the circle in a coordinate plane. In this article, we will discuss how to find the equation of a circle given its center and a point that lies on the circle.
The General Equation of a Circle
The general 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.
Finding the Equation of a Circle
To find the equation of a circle, we need to know its center and a point that lies on the circle. In this problem, we are given the center of the circle as (5, -4) and a point that lies on the circle as (-3, 2).
Step 1: Find the Distance Between the Center and the Point
To find the equation of the circle, we need to find the distance between the center and the point. This distance is equal to the radius of the circle.
The distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the distance between the center (5, -4) and the point (-3, 2) is:
d = sqrt((-3 - 5)^2 + (2 - (-4))^2) = sqrt((-8)^2 + (6)^2) = sqrt(64 + 36) = sqrt(100) = 10
Step 2: Write the Equation of the Circle
Now that we have found the radius of the circle, we can write the equation of the circle using the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center of the circle is (5, -4), and the radius is 10. Therefore, the equation of the circle is:
(x - 5)^2 + (y + 4)^2 = 100
Conclusion
In this article, we discussed how to find the equation of a circle given its center and a point that lies on the circle. We used the general equation of a circle and the distance formula to find the equation of the circle. The equation of the circle is (x - 5)^2 + (y + 4)^2 = 100.
Example Problems
- A circle is centered at the point (2, 3) and passes through the point (5, 1). Find the equation of the circle.
- A circle is centered at the point (-2, 4) and passes through the point (1, -2). Find the equation of the circle.
Solutions
- To find the equation of the circle, we need to find the distance between the center (2, 3) and the point (5, 1). This distance is:
d = sqrt((5 - 2)^2 + (1 - 3)^2) = sqrt((3)^2 + (-2)^2) = sqrt(9 + 4) = sqrt(13)
The radius of the circle is sqrt(13). Therefore, the equation of the circle is:
(x - 2)^2 + (y - 3)^2 = 13
- To find the equation of the circle, we need to find the distance between the center (-2, 4) and the point (1, -2). This distance is:
d = sqrt((1 - (-2))^2 + (-2 - 4)^2) = sqrt((3)^2 + (-6)^2) = sqrt(9 + 36) = sqrt(45)
The radius of the circle is sqrt(45). Therefore, the equation of the circle is:
(x + 2)^2 + (y - 4)^2 = 45
Key Takeaways
- The general equation of a circle is (x - h)^2 + (y - k)^2 = r^2.
- To find the equation of a circle, we need to know its center and a point that lies on the circle.
- The distance between the center and the point is equal to the radius of the circle.
- We can use the distance formula to find the distance between two points.
- The equation of a circle can be written using the general equation of a circle and the distance between the center and the point.
Frequently Asked Questions (FAQs) About Circles =====================================================
Q: What is the center of a circle?
A: The center of a circle is the point that is equidistant from all points on the circle. It is the point around which the circle is centered.
Q: How do I find the center of a circle?
A: To find the center of a circle, you need to know the equation of the circle. The center of the circle is the point (h, k) in the equation (x - h)^2 + (y - k)^2 = r^2.
Q: What is the radius of a circle?
A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is a measure of the size of the circle.
Q: How do I find the radius of a circle?
A: To find the radius of a circle, you need to know the equation of the circle. The radius of the circle is the value of r in the equation (x - h)^2 + (y - k)^2 = r^2.
Q: What is the equation of a circle?
A: The equation of a circle is a mathematical representation of the circle in the form (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 write the equation of a circle?
A: To write the equation of a circle, you need to know the center and radius of the circle. You can use the general equation of a circle (x - h)^2 + (y - k)^2 = r^2 and plug in the values of h, k, and r.
Q: What is the difference between a circle and an ellipse?
A: A circle is a set of points that are equidistant from a central point, while an ellipse is a set of points that are equidistant from two central points. In other words, a circle is a special type of ellipse where the two central points are the same.
Q: How do I find the equation of an ellipse?
A: To find the equation of an ellipse, you need to know the center and the lengths of the semi-major and semi-minor axes. The equation of an ellipse is in the form ((x - h)2/a2) + ((y - k)2/b2) = 1, where (h, k) is the center of the ellipse, a is the length of the semi-major axis, and b is the length of the semi-minor axis.
Q: What is the relationship between a circle and a sphere?
A: A circle is a two-dimensional shape, while a sphere is a three-dimensional shape. A sphere is a set of points that are equidistant from a central point in three-dimensional space.
Q: How do I find the equation of a sphere?
A: To find the equation of a sphere, you need to know the center and the radius of the sphere. The equation of a sphere is in the form (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) is the center of the sphere and r is the radius of the sphere.
Q: What is the difference between a circle and a ring?
A: A circle is a set of points that are equidistant from a central point, while a ring is a set of points that are equidistant from a central point and have a specific width or thickness.
Q: How do I find the equation of a ring?
A: To find the equation of a ring, you need to know the center, radius, and width of the ring. The equation of a ring is in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the ring, r is the radius of the ring, and w is the width of the ring.
Conclusion
In this article, we have discussed some frequently asked questions about circles, including the center, radius, equation, and relationship to other shapes. We have also provided examples and explanations to help you understand these concepts.