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 (1)
The radius of the circle is calculated as half of the diameter, which is 6.
The general equation of a circle with center ( h , 0 ) on the x-axis and radius 6 is ( x − h ) 2 + y 2 = 36 .
Possible equations are derived by considering different values for h (e.g., h = 6 and h = − 6 ).
The equations that match the given options are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 .
( x − 6 ) 2 + y 2 = 36 , ( x + 6 ) 2 + y 2 = 36
Explanation
Analyze the problem and available data. The diameter of the circle is 12 units, so the radius is half of that, which is 6 units. Since the center of the circle lies on the x-axis, its y-coordinate is 0. The general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius. In this case, k = 0 and r = 6 , so the equation becomes ( x − h ) 2 + y 2 = 6 2 = 36 . We need to find the possible values of h and see which of the given equations match.
Consider the case where h=0. If the center is at the origin, then h = 0 , and the equation is x 2 + y 2 = 36 .
Consider the case where h=6. If the center is at ( 6 , 0 ) , then h = 6 , and the equation is ( x − 6 ) 2 + y 2 = 36 .
Consider the case where h=-6. If the center is at ( − 6 , 0 ) , then h = − 6 , and the equation is ( x + 6 ) 2 + y 2 = 36 .
Check the given equations. Now, let's check which of the given equations match our possibilities:
( x − 12 ) 2 + y 2 = 12 : This does not match any of our derived equations.
( x − 6 ) 2 + y 2 = 36 : This matches the case where h = 6 .
x 2 + y 2 = 12 : This does not match any of our derived equations.
x 2 + y 2 = 144 : This does not match any of our derived equations.
( x + 6 ) 2 + y 2 = 36 : This matches the case where h = − 6 .
( x + 12 ) 2 + y 2 = 144 : This does not match any of our derived equations.
Final Answer. Therefore, the possible equations for 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 sprinklers to ensure complete coverage. If the garden has a radius of 6 meters and the center is 6 meters to the right of the origin, the equation ( x − 6 ) 2 + y 2 = 36 helps in accurately mapping the garden's boundaries and planning the irrigation system effectively. Similarly, in architecture, circular windows or domes can be precisely designed using these equations, ensuring structural integrity and aesthetic appeal.
