Write The Equation In Standard Form. Identify The Center And Radius.Equation: X 2 + Y 2 − 10 X − 6 Y + 18 = 0 X^2 + Y^2 - 10x - 6y + 18 = 0 X 2 + Y 2 − 10 X − 6 Y + 18 = 0 Standard Form: \qquad Center: \qquad Radius: \qquad

Introduction

In mathematics, the equation of a circle is a fundamental concept that is used to represent a circle on a coordinate plane. 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. In this article, we will learn how to convert the given equation of a circle to standard form and identify its center and radius.

The Given Equation

The given equation of a circle is $x^2 + y^2 - 10x - 6y + 18 = 0$. To convert this equation to standard 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 and subtract $(10/2)^2 = 25$ inside the parentheses. This will give us $(x - 5)^2 - 25$.

Completing the Square for the $y$ Terms

To complete the square for the $y$ terms, we need to add and subtract $(6/2)^2 = 9$ inside the parentheses. This will give us $(y - 3)^2 - 9$.

Rewriting the Equation in Standard Form

Now that we have completed the square for both the $x$ and $y$ terms, we can rewrite the equation in standard form. We have:

$x^2 + y^2 - 10x - 6y + 18 = 0$

$(x - 5)^2 - 25 + (y - 3)^2 - 9 + 18 = 0$

$(x - 5)^2 + (y - 3)^2 - 16 = 0$

$(x - 5)^2 + (y - 3)^2 = 16$

Identifying the Center and Radius

Now that we have rewritten the equation in standard form, we can identify the center and radius of the circle. The center of the circle is given by $(h, k) = (5, 3)$, and the radius is given by $r = \sqrt{16} = 4$.

Conclusion

In this article, we learned how to convert the given equation of a circle to standard form and identify its center and radius. We completed the square for both the $x$ and $y$ terms and rewrote the equation in standard form. We then identified the center and radius of the circle, which are given by $(h, k) = (5, 3)$ and $r = 4$, respectively.

Example Problems

Problem 1

Write the equation of a circle in standard form, given that the center is $(2, 4)$ and the radius is $3$.

Solution

The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$. Substituting the values of $h = 2$, $k = 4$, and $r = 3$, we get:

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

$(x - 2)^2 + (y - 4)^2 = 9$

Problem 2

Write the equation of a circle in standard form, given that the center is $(-3, 2)$ and the radius is $5$.

Solution

The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$. Substituting the values of $h = -3$, $k = 2$, and $r = 5$, we get:

$(x + 3)^2 + (y - 2)^2 = 5^2$

$(x + 3)^2 + (y - 2)^2 = 25$

Tips and Tricks

• To convert the equation of a circle to standard form, you need to complete the square for both the $x$ and $y$ terms.
• The center of the circle is given by $(h, k)$, and the radius is given by $r = \sqrt{r^2}$.
• You can use the equation of a circle in standard form to find the center and radius of the circle.

Conclusion

In this article, we learned how to convert the given equation of a circle to standard form and identify its center and radius. We completed the square for both the $x$ and $y$ terms and rewrote the equation in standard form. We then identified the center and radius of the circle, which are given by $(h, k) = (5, 3)$ and $r = 4$, respectively. We also provided example problems and tips and tricks to help you master this concept.

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: How do I convert the equation of a circle to standard form?

A: To convert the equation of a circle to standard form, you need to complete the square for both the $x$ and $y$ terms. This involves adding and subtracting the square of half the coefficient of the $x$ or $y$ term inside the parentheses.

Q: What is the center of the circle?

A: The center of the circle is given by $(h, k)$, where $h$ is the $x$-coordinate of the center and $k$ is the $y$-coordinate of the center.

Q: How do I find the center of the circle?

A: To find the center of the circle, you need to rewrite the equation of the circle in standard form. The center is then given by the values of $h$ and $k$ in the standard form equation.

Q: What is the radius of the circle?

A: The radius of the circle is given by $r = \sqrt{r^2}$, where $r^2$ is the value on the right-hand side of the standard form equation.

Q: How do I find the radius of the circle?

A: To find the radius of the circle, you need to rewrite the equation of the circle in standard form. The radius is then given by the square root of the value on the right-hand side of the standard form equation.

Q: Can I use the equation of a circle in standard form to find the center and radius of the circle?

A: Yes, you can use the equation of a circle in standard form to find the center and radius of the circle. The center is given by $(h, k)$, and the radius is given by $r = \sqrt{r^2}$.

Q: What are some common mistakes to avoid when converting the equation of a circle to standard form?

A: Some common mistakes to avoid when converting the equation of a circle to standard form include:

• Not completing the square for both the $x$ and $y$ terms
• Not adding and subtracting the square of half the coefficient of the $x$ or $y$ term inside the parentheses
• Not rewriting the equation in standard form
• Not identifying the center and radius of the circle correctly

Q: How can I practice converting the equation of a circle to standard form and identifying its center and radius?

A: You can practice converting the equation of a circle to standard form and identifying its center and radius by working through example problems and exercises. You can also use online resources and practice tests to help you master this concept.

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

A: The equation of a circle has many real-world applications, including:

• Modeling the path of a projectile
• Describing the shape of a circle in a coordinate plane
• Finding the center and radius of a circle
• Calculating the area and circumference of a circle

Q: Can I use the equation of a circle in standard form to solve problems involving circles?

A: Yes, you can use the equation of a circle in standard form to solve problems involving circles. The equation of a circle in standard form can be used to find the center and radius of a circle, as well as to calculate the area and circumference of a circle.