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 $x^2+y^2=9$.
Answer (2)
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 , T r u e , 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. We will rewrite the equation in standard form to identify the circle's center and radius, and then evaluate each statement.
Complete the square To rewrite the equation in standard form, we complete the square for the x terms. We have 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, which is ( − 1 ) 2 = 1 . So we can rewrite x 2 − 2 x as ( x − 1 ) 2 − 1 .
Rewrite in standard form Substituting this back into the original equation, we get ( x − 1 ) 2 − 1 + y 2 − 8 = 0 . Simplifying, we have ( x − 1 ) 2 + y 2 = 9 . This is the standard form of the equation of a circle with center ( 1 , 0 ) and radius 9 = 3 .
Evaluate each statement Now we evaluate each statement:
The radius of the circle is 3 units. This is true, as we found the radius to be 3.
The center of the circle lies on the x -axis. This is true, since the center is ( 1 , 0 ) , and the y -coordinate is 0.
The center of the circle lies on the y -axis. This is false, since the center is ( 1 , 0 ) , and the x -coordinate is not 0.
The standard form of the equation is ( x − 1 ) 2 + y 2 = 3 . This is false, as the standard form is ( x − 1 ) 2 + y 2 = 9 .
The radius of this circle is the same as the radius of the circle whose equation is x 2 + y 2 = 9 . This is true, since the radius of x 2 + y 2 = 9 is 9 = 3 , which is the same as the radius of the given circle.
Identify the true statements Therefore, 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 the properties of circles, such as their radius and center, is crucial in various real-world applications. For instance, when designing a circular garden, knowing the radius helps determine the amount of fencing needed. Similarly, in architecture, the equation of a circle can be used to plan the layout of circular structures, ensuring precise dimensions and optimal use of space. These concepts also extend to fields like astronomy, where understanding circular orbits is fundamental.
The true statements about the circle defined by the equation x 2 + y 2 − 2 x − 8 = 0 are: the radius is 3 units, the center lies on the x-axis, and the radius is the same as that of the circle given by x 2 + y 2 = 9 .
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