A circle has a diameter of 12 units, and its center lies on the $x$-axis. What could be the equation of the circle? Check all that apply.
$(x-12)^2+y^2=12$
$(x-6)^2+y^2=36$
$x^2+y^2=12$
$x^2+y^2=144$
$(x+6)^2+y^2=36$
$(x+12)^2+y^2=144
Answer (2)
The radius of the circle is calculated as half of the diameter, which is 6 units.
The general equation of a circle with center ( h , 0 ) on the x-axis and radius 6 is ( x − h ) 2 + y 2 = 36 .
By checking the given options, the equations that fit this form are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 .
The possible equations for the circle are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 .
Explanation
Problem Analysis The problem states that a circle has a diameter of 12 units and its center lies on the x-axis. We need to identify the possible equations of the circle from the given options.
Calculate the radius The diameter of the circle is 12 units, so the radius is half of the diameter, which is 6 units. Therefore, the radius squared, r 2 , is 6 2 = 36 .
General equation of the circle Since the center of the circle lies on the x-axis, its y-coordinate is 0. Let the x-coordinate of the center be h . Thus, the center of the circle is ( h , 0 ) . The general equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . In our case, k = 0 and r 2 = 36 , so the equation becomes ( x − h ) 2 + y 2 = 36 .
Check each option Now, we will check each of the given equations to see if they fit the form ( x − h ) 2 + y 2 = 36 .
( x − 12 ) 2 + y 2 = 12 : This equation has a center at (12, 0) but the radius squared is 12, not 36. So, this is incorrect.
( x − 6 ) 2 + y 2 = 36 : This equation has a center at (6, 0) and the radius squared is 36. So, this is a possible equation.
x 2 + y 2 = 12 : This equation has a center at (0, 0) but the radius squared is 12, not 36. So, this is incorrect.
x 2 + y 2 = 144 : This equation has a center at (0, 0) and the radius squared is 144, not 36. So, this is incorrect.
( x + 6 ) 2 + y 2 = 36 : This equation has a center at (-6, 0) and the radius squared is 36. So, this is a possible equation.
( x + 12 ) 2 + y 2 = 144 : This equation has a center at (-12, 0) and the radius squared is 144, not 36. So, this is incorrect.
Final Answer Therefore, the possible equations of the circle are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, knowing the equation helps determine the placement of the sprinkler system to ensure complete coverage. Similarly, in architecture, the equation of a circle is used to create arches and domes, ensuring structural integrity and aesthetic appeal. In navigation, the equation of a circle can define the range of a radar system or the coverage area of a cell tower.
The equations of the circle with a diameter of 12 units are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 . These equations are valid because they have a radius of 6, which is derived from the given diameter. The center of the circle lies on the x -axis.
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