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 in standard form by completing the square: ( x − 1 ) 2 + y 2 = 9 .
Identify the center ( 1 , 0 ) and radius r = 3 from the standard form.
Determine that the radius is 3, the center lies on the x-axis, and the radius is the same as the circle x 2 + y 2 = 9 .
Conclude that the true statements are: radius is 3, center lies on the x-axis, and the radius is the same as the circle x 2 + y 2 = 9 , which can be written as r a d i u s i s 3 , ce n t er l i es o n x − a x i s , s am e r a d i u s a s x 2 + y 2 = 9 .
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 and need to determine which of the given statements are true. To do this, we will first 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 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, which gives ( − 1 ) 2 = 1 . Thus, we can rewrite x 2 − 2 x as ( x − 1 ) 2 − 1 .
Substitute back into the original equation. 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 . Now the equation is in standard form.
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 9 = 3 .
Evaluate each statement. Now we evaluate each statement:
Statement 1: The radius of the circle is 3 units. The radius we found is 3, so this statement is true.
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 statement 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 statement is false.
Statement 4: The standard form of the equation is ( x − 1 ) 2 + y 2 = 3 . The standard form we found is ( x − 1 ) 2 + y 2 = 9 , not 3. This statement 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 the circle x 2 + y 2 = 9 is 9 = 3 . The radius of our circle is also 3. Thus, the radii are the same, and this statement is true.
List 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 essential in many real-world applications. For instance, engineers use the properties of circles to design gears and wheels. Architects rely on circular geometry to create domes and arches. Even in fields like astronomy, understanding the equations of circles helps in modeling the orbits of planets and satellites. By mastering the concepts of circles, you're equipped to tackle a wide range of practical problems in various disciplines.
