Write The Standard Form Of The Equation Of The Circle With The Given Center And Radius. Center: { (2,3)$}$, { R=7$} . T Y P E T H E S T A N D A R D F O R M O F T H E E Q U A T I O N O F T H E C I R C L E . .Type The Standard Form Of The Equation Of The Circle. . T Y P E T H Es T An D A R Df Or M O F T H Ee Q U A T I O N O F T H Ec I Rc L E . { \square\$} (Simplify Your Answer.)
Introduction
In mathematics, a circle is a set of points that are equidistant from a central point called the center. The standard form of the equation of a circle is a mathematical representation of this concept. It is used to describe the shape and size of a circle in terms of its center and radius. In this article, we will discuss how to write the standard form of the equation of a circle with a given center and radius.
What is the Standard Form of the Equation of a Circle?
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.
Understanding the Components of the Standard Form
- (x - h): This represents the horizontal distance between a point on the circle and the center of the circle.
- (y - k): This represents the vertical distance between a point on the circle and the center of the circle.
- r^2: This represents the square of the radius of the circle.
Writing the Standard Form of the Equation of a Circle
To write the standard form of the equation of a circle, we need to know the center and radius of the circle. Let's consider an example where the center of the circle is (2, 3) and the radius is 7.
Step 1: Identify the Center and Radius
The center of the circle is (2, 3) and the radius is 7.
Step 2: Plug in the Values into the Standard Form
Substitute the values of the center and radius into the standard form of the equation of a circle:
(x - 2)^2 + (y - 3)^2 = 7^2
Step 3: Simplify the Equation
Simplify the equation by evaluating the square of the radius:
(x - 2)^2 + (y - 3)^2 = 49
Conclusion
In this article, we discussed how to write the standard form of the equation of a circle with a given center and radius. We used the formula (x - h)^2 + (y - k)^2 = r^2 and plugged in the values of the center and radius to obtain the standard form of the equation of the circle. We also simplified the equation by evaluating the square of the radius.
Example Problems
- Write the standard form of the equation of a circle with a center at (4, 5) and a radius of 10.
- Write the standard form of the equation of a circle with a center at (1, 2) and a radius of 5.
- Write the standard form of the equation of a circle with a center at (3, 4) and a radius of 8.
Answer Key
- (x - 4)^2 + (y - 5)^2 = 100
- (x - 1)^2 + (y - 2)^2 = 25
- (x - 3)^2 + (y - 4)^2 = 64
Frequently Asked Questions (FAQs) about the Standard Form of the Equation of a Circle =====================================================================================
Q: What is the standard 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.
Q: What is the significance of the center and radius in the standard form of the equation of a circle?
A: The center (h, k) represents the point around which the circle is centered, and the radius (r) represents the distance from the center to any point on the circle.
Q: How do I write the standard form of the equation of a circle if I know the center and radius?
A: To write the standard form of the equation of a circle, simply substitute the values of the center and radius into the formula:
(x - h)^2 + (y - k)^2 = r^2
Q: What if I have a circle with a center at (0, 0) and a radius of 5? How do I write the standard form of the equation of this circle?
A: Since the center is at (0, 0), the equation becomes:
x^2 + y^2 = 25
Q: Can I have a circle with a center at (3, 4) and a radius of 2? How do I write the standard form of the equation of this circle?
A: Yes, you can have a circle with a center at (3, 4) and a radius of 2. The standard form of the equation of this circle is:
(x - 3)^2 + (y - 4)^2 = 4
Q: What if I have a circle with a center at (2, 3) and a radius of 7? How do I write the standard form of the equation of this circle?
A: Since the center is at (2, 3) and the radius is 7, the standard form of the equation of this circle is:
(x - 2)^2 + (y - 3)^2 = 49
Q: Can I have a circle with a center at (0, 0) and a radius of 0? How do I write the standard form of the equation of this circle?
A: Yes, you can have a circle with a center at (0, 0) and a radius of 0. However, this is a degenerate case, and the standard form of the equation of this circle is:
x^2 + y^2 = 0
Q: What if I have a circle with a center at (3, 4) and a radius of 0? How do I write the standard form of the equation of this circle?
A: Since the radius is 0, the circle is a single point, and the standard form of the equation of this circle is:
(x - 3)^2 + (y - 4)^2 = 0
Conclusion
In this article, we answered some frequently asked questions about the standard form of the equation of a circle. We discussed the significance of the center and radius, how to write the standard form of the equation of a circle, and some examples of circles with different centers and radii. We hope this article has been helpful in understanding the standard form of the equation of a circle.