Identify The Center And Radius Of The Circle.a) $x^2 + (y+1)^2 = 4$

Introduction

In mathematics, a circle is a set of points that are equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this article, we will discuss how to identify the center and radius of a circle given its equation in the standard form.

Standard Form of a Circle Equation

The standard form of a circle equation is:

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

where (h, k) is the center of the circle and r is the radius.

Example: Identifying the Center and Radius of a Circle

Let's consider the equation of a circle:

$x^2 + (y+1)^2 = 4$

To identify the center and radius of this circle, we need to rewrite the equation in the standard form.

Step 1: Expand the Equation

We can start by expanding the squared term:

$x^2 + (y^2 + 2y + 1) = 4$

Step 2: Combine Like Terms

Next, we can combine like terms:

$x^2 + y^2 + 2y + 1 = 4$

Step 3: Rearrange the Equation

Now, we can rearrange the equation to group the x-terms and y-terms:

$x^2 + y^2 + 2y = 3$

Step 4: Complete the Square

To complete the square, we need to add and subtract the square of half the coefficient of the y-term:

$x^2 + (y^2 + 2y + 1) = 3 + 1$

$x^2 + (y+1)^2 = 4$

Step 5: Identify the Center and Radius

Now that we have the equation in the standard form, we can identify the center and radius of the circle:

• The center of the circle is at (h, k) = (0, -1)
• The radius of the circle is r = √4 = 2

Conclusion

In this article, we discussed how to identify the center and radius of a circle given its equation in the standard form. We used the equation $x^2 + (y+1)^2 = 4$ as an example and walked through the steps to rewrite the equation in the standard form. We then identified the center and radius of the circle as (0, -1) and 2, respectively.

Key Takeaways

• The standard form of a circle equation is (x - h)^2 + (y - k)^2 = r^2
• To identify the center and radius of a circle, we need to rewrite the equation in the standard form
• We can complete the square to rewrite the equation in the standard form
• The center of the circle is at (h, k) and the radius is r

Frequently Asked Questions

Q: What is the standard form of a circle equation?

A: The standard form of a circle equation is (x - h)^2 + (y - k)^2 = r^2.

Q: How do I identify the center and radius of a circle?

A: To identify the center and radius of a circle, we need to rewrite the equation in the standard form by completing the square.

Q: What is the center of the circle in the equation $x^2 + (y+1)^2 = 4$?

A: The center of the circle is at (0, -1).

Q: What is the radius of the circle in the equation $x^2 + (y+1)^2 = 4$?

Introduction

In our previous article, we discussed how to identify the center and radius of a circle given its equation in the standard form. In this article, we will answer some frequently asked questions about circle equations.

Q: What is the standard form of a circle equation?

A: The standard form of a circle equation is:

(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 identify the center and radius of a circle?

A: To identify the center and radius of a circle, we need to rewrite the equation in the standard form by completing the square.

Q: What is the center of the circle in the equation $x^2 + (y+1)^2 = 4$?

A: The center of the circle is at (0, -1).

Q: What is the radius of the circle in the equation $x^2 + (y+1)^2 = 4$?

A: The radius of the circle is 2.

Q: How do I find the equation of a circle with a given center and radius?

A: To find the equation of a circle with a given center and radius, we can use the standard form of a circle equation:

(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 equation of a circle with a center at (2, 3) and a radius of 4?

A: The equation of a circle with a center at (2, 3) and a radius of 4 is:

(x - 2)^2 + (y - 3)^2 = 16

Q: How do I find the distance between the center of a circle and a point on the circle?

A: To find the distance between the center of a circle and a point on the circle, we can use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) is the center of the circle and (x2, y2) is the point on the circle.

Q: What is the distance between the center of the circle $x^2 + (y+1)^2 = 4$ and the point (1, 0)?

A: The distance between the center of the circle $x^2 + (y+1)^2 = 4$ and the point (1, 0) is:

d = √((1 - 0)^2 + (0 - (-1))^2) = √(1 + 1) = √2

Q: How do I find the equation of a circle that passes through three given points?

A: To find the equation of a circle that passes through three given points, we can use the following steps:

1. Find the center of the circle using the formula:

(h, k) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

2. Find the radius of the circle using the formula:

r = √((x2 - x1)^2 + (y2 - y1)^2)

3. Use the standard form of a circle equation to write the equation of the circle.

Q: What is the equation of a circle that passes through the points (1, 2), (3, 4), and (5, 6)?

A: The equation of a circle that passes through the points (1, 2), (3, 4), and (5, 6) is:

(x - 3)^2 + (y - 4)^2 = 10

Conclusion

In this article, we answered some frequently asked questions about circle equations. We discussed how to identify the center and radius of a circle, how to find the equation of a circle with a given center and radius, and how to find the distance between the center of a circle and a point on the circle. We also discussed how to find the equation of a circle that passes through three given points.

Key Takeaways

• The standard form of a circle equation is (x - h)^2 + (y - k)^2 = r^2
• To identify the center and radius of a circle, we need to rewrite the equation in the standard form by completing the square
• We can find the equation of a circle with a given center and radius using the standard form of a circle equation
• We can find the distance between the center of a circle and a point on the circle using the distance formula
• We can find the equation of a circle that passes through three given points using the following steps: find the center of the circle, find the radius of the circle, and use the standard form of a circle equation to write the equation of the circle.