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 center and radius.
Determine the true statements: 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 . Statements 1, 2, and 5 are true.
Explanation
Analyze the problem and rewrite the equation in standard form. 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, which is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center of the circle and r is the radius.
Complete the square for x terms. To rewrite the given equation in standard form, we need to 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 , 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 .
Substitute and simplify the equation. Now, substitute this back into the original equation: ( x − 1 ) 2 − 1 + y 2 − 8 = 0 .
Simplify the equation: ( x − 1 ) 2 + y 2 = 9 .
This is the standard form of the equation of the circle.
Identify the center and radius. From the standard form ( x − 1 ) 2 + y 2 = 9 , we can identify the center of the circle as ( 1 , 0 ) and the radius as r = 9 = 3 .
Evaluate the given statements. Now we can evaluate the given statements:
The radius of the circle is 3 units. This is true because we found that r = 3 .
The center of the circle lies on the x -axis. This is true because the center is ( 1 , 0 ) , and the y -coordinate is 0.
The center of the circle lies on the y -axis. This is false because 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 because 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 because the radius of x 2 + y 2 = 9 is 9 = 3 , which is the same as the radius of the given circle.
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. Imagine designing a circular tunnel; knowing the equation of the circle helps determine the tunnel's dimensions and ensures structural integrity. Similarly, in architecture, circular designs are common in domes and arches, where understanding the circle's properties is essential for stability and aesthetics. Even in fields like astronomy, understanding circular orbits is fundamental to predicting the movement of celestial bodies.
The true statements are that the radius of the circle is 3 units, the center lies on the x-axis, and the radius is the same as that of the circle whose equation is x 2 + y 2 = 9 .
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