Consider A Circle Whose Equation Is $x^2 + Y^2 - 2x - 8 = 0$. Which Statements Are True? Select Three Options.A. The Radius Of The Circle Is 3 Units.B. The Center Of The Circle Lies On The $x$-axis.C. The Center Of The Circle Lies On

In mathematics, the equation of a circle is a fundamental concept that is used to describe the shape and position of a circle in a two-dimensional plane. The general equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the coordinates of the center of the circle and $r$ represents the radius of the circle.

Given Equation and Analysis

The given equation of the circle is $x^2 + y^2 - 2x - 8 = 0$. To analyze this equation, we need to rewrite it in the standard form of the equation of a circle. We can do this by completing the square for both the $x$ and $y$ terms.

Completing the Square for the x Term

To complete the square for the $x$ term, we need to add and subtract $(2/2)^2 = 1$ inside the parentheses.

x^2 - 2x + 1 - 1 + y^2 - 8 = 0

Completing the Square for the y Term

To complete the square for the $y$ term, we need to add and subtract $(0/2)^2 = 0$ inside the parentheses.

(x - 1)^2 - 1 + y^2 - 8 = 0

Simplifying the Equation

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

(x - 1)^2 + y^2 - 9 = 0

Standard Form of the Equation

The standard form of the equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$. Comparing this with the simplified equation, we can see that the center of the circle is at $(1, 0)$ and the radius is $\sqrt{9} = 3$ units.

Analyzing the Options

Now, let's analyze the given options.

Option A: The radius of the circle is 3 units.

As we have already seen, the radius of the circle is indeed 3 units. Therefore, this statement is true.

Option B: The center of the circle lies on the x-axis.

The center of the circle is at $(1, 0)$, which lies on the x-axis. Therefore, this statement is true.

Option C: The center of the circle lies on the y-axis.

The center of the circle is at $(1, 0)$, which does not lie on the y-axis. Therefore, this statement is false.

Conclusion

In conclusion, the statements that are true are:

• The radius of the circle is 3 units.
• The center of the circle lies on the x-axis.

The statement that is false is:

• The center of the circle lies on the y-axis.

Key Takeaways

• The equation of a circle can be rewritten in the standard form by completing the square for both the x and y terms.
• The center of the circle can be found by comparing the rewritten equation with the standard form of the equation of a circle.
• The radius of the circle can be found by taking the square root of the constant term on the right-hand side of the equation.
  Frequently Asked Questions (FAQs) about Circles =====================================================

In this article, we will answer some frequently asked questions about circles, including their equation, properties, and applications.

Q: What is the equation of a circle?

A: The equation of a circle is a mathematical expression that describes the shape and position of a circle in a two-dimensional plane. The general equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ represents the coordinates of the center of the circle and $r$ represents the radius of the circle.

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

A: To find the center of a circle, you need to rewrite the equation of the circle in the standard form by completing the square for both the x and y terms. The center of the circle is then given by the coordinates $(h, k)$.

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

A: To find the radius of a circle, you need to take the square root of the constant term on the right-hand side of the equation. The radius of the circle is then given by $r = \sqrt{r^2}$.

Q: What is the difference between a circle and a sphere?

A: A circle is a two-dimensional shape that lies in a plane, while a sphere is a three-dimensional shape that is a ball. The equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$, while the equation of a sphere is given by $(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$, where $(h, k, l)$ represents the coordinates of the center of the sphere.

Q: What is the circumference of a circle?

A: The circumference of a circle is the distance around the circle. It is given by the formula $C = 2\pi r$, where $r$ is the radius of the circle.

Q: What is the area of a circle?

A: The area of a circle is the amount of space inside the circle. It is given by the formula $A = \pi r^2$, where $r$ is the radius of the circle.

Q: What are some real-world applications of circles?

A: Circles have many real-world applications, including:

• Geometry and trigonometry: Circles are used to describe the shape and position of objects in space.
• Physics and engineering: Circles are used to describe the motion of objects and the behavior of physical systems.
• Computer graphics: Circles are used to create images and animations.
• Architecture: Circles are used to design buildings and other structures.

Conclusion

In conclusion, circles are an important concept in mathematics and have many real-world applications. By understanding the equation and properties of a circle, you can solve problems and create new things.

Key Takeaways

• The equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$.
• The center of a circle is given by the coordinates $(h, k)$.
• The radius of a circle is given by $r = \sqrt{r^2}$.
• The circumference of a circle is given by $C = 2\pi r$.
• The area of a circle is given by $A = \pi r^2$.