Consider a circle whose equation is $x^2+y^2-2 x-8=0$. 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 $x$-axis.
C. The center of the circle lies on the $y$-axis.
D. The standard form of the equation is $(x-1)^2+y^2=3$.
E. The radius of this circle is the same as the radius of the circle whose equation is $x^2+y^2=9$.

The true statements about the circle are: the radius of the circle is 3 units, the center lies on the x-axis, and the radius is the same as the circle defined by the equation x 2 + y 2 = 9 .
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Answered by Anonymous | 2025-07-05

Rewrite the given circle equation x 2 + y 2 − 2 x − 8 = 0 in standard form by completing the square: ( x − 1 ) 2 + y 2 = 9 .
Identify the center and radius from the standard form: center ( 1 , 0 ) , radius r = 3 .
Evaluate each statement based on the derived center and radius.
The true statements are: radius is 3, center lies on the x-axis, and the radius is the same as that of the circle x 2 + y 2 = 9 . T r u e ​

Explanation

Analyze the problem We are given the equation of a circle: x 2 + y 2 − 2 x − 8 = 0 . Our goal is to determine which of the given statements about this circle are true. To do this, we will rewrite the equation in standard form, identify the center and radius, and then evaluate each statement.

Complete the square To rewrite the equation in standard form, we need to complete the square for the x terms. The equation is x 2 + y 2 − 2 x − 8 = 0 . We focus on the x terms: x 2 − 2 x . To complete the square, we take half of the coefficient of the x term, which is − 2/2 = − 1 , and square it: ( − 1 ) 2 = 1 . So we add and subtract 1 to complete the square: x 2 − 2 x + 1 − 1 = ( x − 1 ) 2 − 1 .

Rewrite in standard form Now we substitute this back into the original equation: ( x − 1 ) 2 − 1 + y 2 − 8 = 0 . This simplifies to ( x − 1 ) 2 + y 2 = 9 . This is the standard form of the equation of a circle, ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.

Identify center and radius From the standard form ( x − 1 ) 2 + y 2 = 9 , we can identify the center as ( 1 , 0 ) and the radius as r = 9 ​ = 3 .

Evaluate statements Now we evaluate each statement:



Statement 1: The radius of the circle is 3 units. This is true, as we found the radius to be 3.
Statement 2: The center of the circle lies on the x -axis. The center is ( 1 , 0 ) . Since the y -coordinate is 0, the center lies on the x -axis. This is true.
Statement 3: The center of the circle lies on the y -axis. The center is ( 1 , 0 ) . Since the x -coordinate is not 0, the center does not lie on the y -axis. This is false.
Statement 4: The standard form of the equation is ( x − 1 ) 2 + y 2 = 3 . The correct standard form is ( x − 1 ) 2 + y 2 = 9 . This is false.
Statement 5: The radius of this circle is the same as the radius of the circle whose equation is x 2 + y 2 = 9 . The radius of x 2 + y 2 = 9 is 9 ​ = 3 , which is the same as the radius of our circle. This is true.


Final Answer The true statements are:


The radius of the circle is 3 units.
The center of the circle lies on the x -axis.
The radius of this circle is the same as the radius of the circle whose equation is x 2 + y 2 = 9 .

Examples
Understanding circles is crucial in many real-world applications. For example, engineers use the properties of circles to design gears and wheels. Architects use circles in building designs, such as domes and arches. In sports, the dimensions of a basketball court include circles, and understanding their properties helps players strategize. Even in astronomy, the orbits of planets are often approximated as circles or ellipses, where understanding the radius and center helps predict planetary positions.

Answered by GinnyAnswer | 2025-07-05