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 (1)
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 as 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 , so half of it is − 1 . Then we square it: ( − 1 ) 2 = 1 . So we can rewrite x 2 − 2 x as ( 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 . Simplifying, we get ( 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 9 = 3 .
Evaluate each statement 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 1 (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 . This is false. The standard form is ( x − 1 ) 2 + y 2 = 9 .
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 the circle x 2 + y 2 = 9 is 9 = 3 , which is the same as the radius of our circle. This is true.
State 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 circles is crucial in many real-world applications. For example, civil engineers use the properties of circles when designing tunnels and bridges. Architects use circles in building designs, and even in creating aesthetically pleasing layouts for rooms and gardens. Knowing how to find the center and radius of a circle from its equation allows engineers and designers to accurately plan and construct circular structures, ensuring stability and visual appeal. Another example is in GPS technology, where your location is determined by finding the intersection of circles from multiple satellites.
