What Is The Radius Of A Circle Whose Equation Is 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 ?A. 2 Units B. 4 Units C. 8 Units D. 16 Units

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Introduction to Circle Equations

A circle is a set of points in a plane that are equidistant from a central point known as the center. The equation of a circle in the standard form 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. However, not all circle equations are in the standard form. In this article, we will explore how to find the radius of a circle given its equation in the general form.

Converting the Equation to Standard Form

The given equation is $x^2 + y^2 - 10x + 6y + 18 = 0$. To convert this equation to the standard form, we need to complete the square for both the $x$ and $y$ terms. We start by grouping the $x$ terms and $y$ terms separately:

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

Next, we add the square of half the coefficient of $x$ to the $x$ terms and the square of half the coefficient of $y$ to the $y$ terms. The coefficient of $x$ is $-10$, so half of it is $-5$, and its square is $25$. The coefficient of $y$ is $6$, so half of it is $3$, and its square is $9$. We add these values to both sides of the equation:

$$(x^2 - 10x + 25) + (y^2 + 6y + 9) = -18 + 25 + 9$$

This simplifies to:

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

Finding the Radius

Now that we have the equation in the standard form, we can easily identify the radius. The radius is the square root of the value on the right-hand side of the equation. In this case, the value is $16$, so the radius is:

$$r = \sqrt{16} = 4$$

Conclusion

In this article, we explored how to find the radius of a circle given its equation in the general form. We converted the given equation to the standard form by completing the square for both the $x$ and $y$ terms. Once we had the equation in the standard form, we were able to easily identify the radius as the square root of the value on the right-hand side of the equation. The radius of the circle is $4$ units.

Frequently Asked Questions

• What is the equation of a circle in the standard form?
  • The equation of a circle in the standard form 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.
• How do I convert a circle equation from the general form to the standard form?
  • To convert a circle equation from the general form to the standard form, you need to complete the square for both the $x$ and $y$ terms.
• What is the radius of a circle with the equation $(x - 5)^2 + ( + 3)^2 = 16$?
  • The radius of a circle with the equation $(x - 5)^2 + (y + 3)^2 = 16$ is $4$ units.

Step-by-Step Solution

1. Start by grouping the $x$ terms and $y$ terms separately.
2. Add the square of half the coefficient of $x$ to the $x$ terms and the square of half the coefficient of $y$ to the $y$ terms.
3. Simplify the equation to get it in the standard form.
4. Identify the radius as the square root of the value on the right-hand side of the equation.

Example Problems

• Find the radius of a circle with the equation $x^2 + y^2 - 12x + 8y + 36 = 0$.
  • To find the radius, we need to convert the equation to the standard form by completing the square for both the $x$ and $y$ terms.
  • Once we have the equation in the standard form, we can easily identify the radius as the square root of the value on the right-hand side of the equation.
• Find the radius of a circle with the equation $(x + 2)^2 + (y - 4)^2 = 25$.
  • The radius of a circle with the equation $(x + 2)^2 + (y - 4)^2 = 25$ is $5$ units.

Key Concepts

• The equation of a circle in the standard form 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.
• To convert a circle equation from the general form to the standard form, you need to complete the square for both the $x$ and $y$ terms.
• The radius of a circle is the square root of the value on the right-hand side of the equation in the standard form.

Summary

In this article, we explored how to find the radius of a circle given its equation in the general form. We converted the given equation to the standard form by completing the square for both the $x$ and $y$ terms. Once we had the equation in the standard form, we were able to easily identify the radius as the square root of the value on the right-hand side of the equation. The radius of the circle is $4$ units.

Q&A: Circle Equations and Radius

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

A: The equation of a circle in the standard form 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 a circle equation from the general form to the standard form?

A: To convert a circle equation from the general form to the standard form, you need to complete the square for both the $x$ and $y$ terms.

Q: What is the radius of a circle with the equation $(x - 5)^2 + (y + 3)^2 = 16$?

A: The radius of a circle with the equation $(x - 5)^2 + (y + 3)^2 = 16$ is $4$ units.

Q: How do I find the center of a circle given its equation in the standard form?

A: To find the center of a circle given its equation in the standard form, you need to identify the values of $h$ and $k$ in the equation $(x - h)^2 + (y - k)^2 = r^2$. The center of the circle is $(h, k)$.

Q: What is the difference between the general form and the standard form of a circle equation?

A: The general form of a circle equation is $x^2 + y^2 + Ax + By + C = 0$, while the standard form is $(x - h)^2 + (y - k)^2 = r^2$. The standard form is more convenient for finding the center and radius of a circle.

Q: Can I find the radius of a circle without converting the equation to the standard form?

A: Yes, you can find the radius of a circle without converting the equation to the standard form. You can use the formula $r = \sqrt{A^2 + B^2 - C}$, where $A$, $B$, and $C$ are the coefficients of the general form.

Q: What is the significance of the center of a circle?

A: The center of a circle is the point that is equidistant from all points on the circle. It is also the point around which the circle is centered.

Q: Can I find the center of a circle without converting the equation to the standard form?

A: Yes, you can find the center of a circle without converting the equation to the standard form. You can use the formula $(h, k) = (-A/2, -B/2)$, where $A$ and $B$ are the coefficients of the general form.

Q: What is the relationship between the radius and the center of a circle?

A: The radius of a circle is the distance from the center to any point on the circle. It is also the square root of the value on the right-hand side of the equation in the standard form.

Q: Can I find the radius of a circle if I know the center and a point on the circle?

A: Yes, you can find the radius of a circle if you know the center and a point on the circle. You can use the formula $r = \sqrt{(x - h)^2 + (y - k)^2}$ where $(h, k)$ is the center and $(x, y)$ is the point.

Q: What is the significance of the radius of a circle?

A: The radius of a circle is the distance from the center to any point on the circle. It is also a measure of the size of the circle.

Q: Can I find the radius of a circle if I know the area of the circle?

A: Yes, you can find the radius of a circle if you know the area of the circle. You can use the formula $r = \sqrt{\frac{A}{\pi}}$, where $A$ is the area of the circle.

Q: What is the relationship between the area and the radius of a circle?

A: The area of a circle is equal to $\pi r^2$, where $r$ is the radius.

Q: Can I find the area of a circle if I know the radius?

A: Yes, you can find the area of a circle if you know the radius. You can use the formula $A = \pi r^2$, where $r$ is the radius.

Q: What is the significance of the area of a circle?

A: The area of a circle is a measure of the size of the circle. It is also used in many real-world applications, such as calculating the area of a room or the surface area of a sphere.

Q: Can I find the circumference of a circle if I know the radius?

A: Yes, you can find the circumference of a circle if you know the radius. You can use the formula $C = 2\pi r$, where $r$ is the radius.

Q: What is the relationship between the circumference and the radius of a circle?

A: The circumference of a circle is equal to $2\pi r$, where $r$ is the radius.

Q: Can I find the radius of a circle if I know the circumference?

A: Yes, you can find the radius of a circle if you know the circumference. You can use the formula $r = \frac{C}{2\pi}$, where $C$ is the circumference.

Q: What is the significance of the circumference of a circle?

A: The circumference of a circle is a measure of the distance around the circle. It is also used in many real-world applications, such as calculating the distance around a room or the perimeter of a shape.

Q: Can I find the diameter of a circle if I know the radius?

A: Yes, you can find the diameter of a circle if you know the radius. You can use the formula $d = 2r$, where $r$ is the radius.

Q: What is the relationship between the diameter and the radius of a circle?

A: The diameter of a circle is equal to $2r$, where $r$ is the radius.

Q: Can I find the radius of a circle if I know the diameter?

A: Yes, you can find the radius of a circle if you know the diameter. You can use the formula $r = \frac{d}{2}$, where $d$ is the diameter.

Q: What is the significance of the diameter of a circle?

A: The diameter of a circle is a measure of the distance across the circle, passing through its center. It is also used in many real-world applications, such as calculating the distance across a or the width of a shape.