Consider A Circle Whose Equation Is X 2 + Y 2 − 2 X − 8 = 0 X^2 + Y^2 - 2x - 8 = 0 X 2 + Y 2 − 2 X − 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 X X -axis. C. The Center Of The Circle Lies

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: $x^2 + y^2 - 2x - 8 = 0$

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

Completing the Square

To complete the square for the $x$ term, we need to add $(2/2)^2 = 1$ to both sides of the equation. Similarly, to complete the square for the $y$ term, we need to add $(0/2)^2 = 0$ to both sides of the equation.

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

Now, we can rewrite the equation as:

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

Understanding the Standard Form

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

Analyzing the Options

Now, let's analyze the given options:

A. The radius of the circle is 3 units.

This statement is true, as we have already determined that the radius of the circle is 3 units.

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

This statement is false, as the center of the circle is at $(1, 0)$, which lies on the $y$-axis, not the $x$-axis.

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

This statement is false, as the center of the circle is at $(1, 0)$, which lies on the $x$-axis, not the $y$-axis.

Conclusion

In conclusion, the correct statements are:

• The radius of the circle is 3 units.
• The center of the circle lies on the $x$-axis (this is incorrect, the center lies on the y-axis).

In the previous article, we discussed the equation of a circle and how to rewrite it in the standard form. We also analyzed the given options and determined that the radius of the circle is 3 units. In this article, we will answer some frequently asked questions about the equation of a circle.

Q: What is the general equation of a circle?

A: 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 rewrite the equation of a circle in the standard form?

A: To rewrite the equation of a circle in the standard 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.

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

A: The center of the circle is the point around which the circle is centered. It is represented by the coordinates $(h, k)$ in the general equation of a circle.

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

A: To find the radius of the circle, you need to take the square root of the value on the right-hand side of the equation. In the standard form of a circle, the radius is represented by $r$.

Q: Can a circle have a negative radius?

A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value.

Q: Can a circle have a zero radius?

A: No, a circle cannot have a zero radius. A circle with a zero radius would be a point, not a circle.

Q: How do I graph a circle?

A: To graph a circle, you need to plot the center of the circle and then draw a circle with the given radius.

Q: What is the equation of a circle with a center at $(0, 0)$ and a radius of 5 units?

A: The equation of a circle with a center at $(0, 0)$ and a radius of 5 units is $x^2 + y^2 = 25$.

Q: What is the equation of a circle with a center at $(3, 4)$ and a radius of 2 units?

A: The equation of a circle with a center at $(3, 4)$ and a radius of 2 units is $(x - 3)^2 + (y - 4)^2 = 4$.

Conclusion

In conclusion, the equation of a circle is a fundamental concept in mathematics that is used to describe the shape and position of a circle in a two-dimensional plane. We have discussed the general equation of a circle, how to rewrite it in the standard form, and answered some frequently asked questions about the equation of a circle.