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 =\ \square$}$.
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 size of the circle. In this article, we will explore 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.
Finding the Center and Radius of the Circle
We are given that the circle is centered at the point (5, -4) and passes through the point (-3, 2). To find the equation of the circle, we need to find the radius.
Step 1: Find the Distance Between the Center and the Point
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to find the distance between the center (5, -4) and the point (-3, 2).
d = √((-3 - 5)^2 + (2 - (-4))^2) d = √((-8)^2 + (6)^2) d = √(64 + 36) d = √100 d = 10
Step 2: Find the Radius of the Circle
The radius of the circle is equal to the distance between the center and the point.
r = d r = 10
Step 3: Write the Equation of the Circle
Now that we have the center (h, k) = (5, -4) and the radius r = 10, we can write the equation of the circle.
(x - 5)^2 + (y + 4)^2 = 100
Conclusion
In this article, we have shown 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 radius of the circle. Finally, we wrote the equation of the circle using the center and radius.
Example Problems
- A circle is centered at the point (2, 3) and passes through the point (6, 1). Find the equation of the circle.
- A circle is centered at the point (-1, 2) and passes through the point (4, -3). Find the equation of the circle.
Answer Key
- (x - 2)^2 + (y - 3)^2 = 25
- (x + 1)^2 + (y - 2)^2 = 25
Discussion
This article has provided a step-by-step guide on 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 and the distance formula were used to find the radius of the circle. The equation of the circle was then written using the center and radius.
Related Topics
- The equation of a circle in standard form
- The equation of a circle in general form
- The distance formula
- The midpoint formula
Glossary
- Center: The point that is equidistant from all points on the circle.
- Radius: The distance between the center and any point on the circle.
- Equation of a circle: A mathematical representation of the shape and size of a circle.
Frequently Asked Questions (FAQs) about the Equation of a Circle ====================================================================
Q: What is the general equation of a circle?
A: 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.
Q: How do I find the center and radius of a circle?
A: To find the center and radius of a circle, you need to know two points that lie on the circle. You can use the distance formula to find the distance between the two points, and then use that distance as the radius.
Q: What is the distance formula?
A: The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the two points.
Q: How do I write the equation of a circle?
A: To write the equation of a circle, you need to know the center (h, k) and the radius r. You can then plug these values into the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
Q: What is the difference between the standard form and general form of the equation of a circle?
A: The standard 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.
The general form of the equation of a circle is given by:
Ax^2 + Ay^2 + Bx + Cy + D = 0
where A, B, C, and D are constants.
Q: How do I convert the general form of the equation of a circle to the standard form?
A: To convert the general form of the equation of a circle to the standard form, you need to complete the square. This involves rearranging the terms and adding and subtracting constants to create a perfect square trinomial.
Q: What is the significance of the center and radius of a circle?
A: The center and radius of a circle are important because they determine the shape and size of the circle. The center is the point that is equidistant from all points on the circle, and the radius is the distance between the center and any point on the circle.
Q: How do I find the equation of a circle given its center and a point that lies on the circle?
A: To find the equation of a circle given its center and a point that lies on the circle, you need to use the general equation of a circle and the distance formula. You can then plug in the values of the center and the point to find the equation of the circle.
Q: What are some common applications of the equation of a circle?
A: The equation of a circle has many common applications in mathematics and science, including:
- Geometry and trigonometry
- Physics and engineering
- Computer graphics and game development
- Statistics and data analysis
Q: How do I use the equation of a circle in real-world problems?
A: The equation of a circle can be used to solve a variety of real-world problems, including:
- Finding the distance between two points
- Calculating the area and circumference of a circle
- Determining the equation of a circle given its center and a point that lies on the circle
- Solving problems involving circles in geometry and trigonometry.