What Is The Equation Of The Circle In Center-radius Form For X 2 + Y 2 + 12 X − 10 Y − 7 = 0 X^2 + Y^2 + 12x - 10y - 7 = 0 X 2 + Y 2 + 12 X − 10 Y − 7 = 0 ?A. ( X − 6 ) 2 + ( Y + 5 ) 2 = 68 (x-6)^2 + (y+5)^2 = 68 ( X − 6 ) 2 + ( Y + 5 ) 2 = 68 B. ( X + 12 ) 2 + ( Y − 25 ) 2 = 68 (x+12)^2 + (y-25)^2 = 68 ( X + 12 ) 2 + ( Y − 25 ) 2 = 68 C. ( X + 6 ) 2 + ( Y − 5 ) 2 = 61 (x+6)^2 + (y-5)^2 = 61 ( X + 6 ) 2 + ( Y − 5 ) 2 = 61 D. ( X + 6 ) 2 + ( Y − 5 ) 2 = 68 (x+6)^2 + (y-5)^2 = 68 ( X + 6 ) 2 + ( Y − 5 ) 2 = 68

Mar 12, 2025 by ADMIN 514 views

**What is the Equation of the Circle in Center-Radius Form?**

Understanding the Center-Radius Form of a Circle

The center-radius form of a circle is a mathematical equation that describes a circle in terms of its center and radius. This form is essential in mathematics, particularly in geometry and algebra. In this article, we will explore how to convert a given circle equation into its center-radius form.

Converting the Given Equation to Center-Radius Form

To convert the given equation $x^2 + y^2 + 12x - 10y - 7 = 0$ into its center-radius form, we need to complete the square for both the x and y terms.

Completing the Square for the x Terms

To complete the square for the x terms, we need to add $(12/2)^2 = 36$ to both sides of the equation.

x^2 + 12x + 36 + y^2 - 10y - 7 = 36

Completing the Square for the y Terms

To complete the square for the y terms, we need to add $(-10/2)^2 = 25$ to both sides of the equation.

x^2 + 12x + 36 + y^2 - 10y + 25 - 7 = 36 + 25

Simplifying the Equation

Now, we can simplify the equation by combining like terms.

(x^2 + 12x + 36) + (y^2 - 10y + 25) = 61

Writing the Equation in Center-Radius Form

Finally, we can write the equation in its center-radius form by factoring the squared terms.

(x + 6)^2 + (y - 5)^2 = 61

Comparing the Result with the Given Options

Now that we have the equation in its center-radius form, we can compare it with the given options.

Option	Equation
A	$(x-6)^2 + (y+5)^2 = 68$
B	$(x+12)^2 + (y-25)^2 = 68$
C	$(x+6)^2 + (y-5)^2 = 61$
D	$(x+6)^2 + (y-5)^2 = 68$

As we can see, the correct equation is option C: $(x+6)^2 + (y-5)^2 = 61$ .

Conclusion

In this article, we learned how to convert a given circle equation into its center-radius form. We completed the square for both the x and y terms and simplified the equation to obtain the center-radius form. We also compared the result with the given options and found that the correct equation is option C: $(x+6)^2 + (y-5)^2 = 61$ .

Frequently Asked Questions

What is the center-radius form of a circle?
How do I convert a given circle equation into its center-radius form?
What is the equation of the circle in center-radius form for $x^2 + y^2 + 12x - 10y - 7 = 0$ ?

Answer

The center-radius form of a circle is a mathematical equation that describes a circle in terms of its center and radius.
To convert a given circle equation into its center-radius form, you need to complete the square for both the x and y terms.
The equation of the circle in center-radius form for $x^2 + y^2 + 12x - 10y - 7 = 0$ is $(x+6)^2 + (y-5)^2 = 61$ .
Frequently Asked Questions: Center-Radius Form of a Circle ===========================================================

Q: What is the center-radius form of a circle?

A: The center-radius form of a circle is a mathematical equation that describes a circle in terms of its center and radius. It is a way to represent a circle using the coordinates of its center and the length of its radius.

Q: How do I convert a given circle equation into its center-radius form?

A: To convert a given circle equation into its center-radius form, you need to complete the square for both the x and y terms. This involves adding and subtracting the same value to both sides of the equation to create a perfect square trinomial.

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

A: The general 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 find the center and radius of a circle from its equation?

A: To find the center and radius of a circle from its equation, you need to rewrite the equation in its center-radius form. Once you have the equation in this form, you can easily identify the center and radius.

Q: What is the difference between the standard form and center-radius form of a circle equation?

A: The standard form of a circle equation is $x^2 + y^2 + Ax + By + C = 0$ , while the center-radius form is $(x-h)^2 + (y-k)^2 = r^2$ . The main difference between the two forms is that the center-radius form explicitly shows the center and radius of the circle.

Q: Can I convert a circle equation from standard form to center-radius form?

A: Yes, you can convert a circle equation from standard form to center-radius form by completing the square for both the x and y terms.

Q: What are some common mistakes to avoid when converting a circle equation to center-radius form?

A: Some common mistakes to avoid when converting a circle equation to center-radius form include:

Not completing the square for both the x and y terms
Adding or subtracting the wrong values to both sides of the equation
Not simplifying the equation after completing the square

Q: How do I check if a given equation is a circle equation?

A: To check if a given equation is a circle equation, you need to see if it can be rewritten in the form $(x-h)^2 + (y-k)^2 = r^2$ . If it can be rewritten in this form, then it is a circle equation.

Q: What are some real-world applications of the center-radius form of a circle equation?

A: Some real-world applications of the center-radius form of a circle equation include:

Calculating the distance between two points on a circle
Finding the area of a circle
Determining the equation of a circle given its center and radius

Conclusion

In this article, we answered some frequently asked questions about the center-radius form of a circle equation. We covered topics such as converting a circle equation to center-radius form, finding the center and radius of a circle, and common mistakes to avoid. We also discussed some real-world applications of the center-radius form of a circle equation.