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 , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . Since the center lies on the x-axis, k = 0 .
Substitute r = 6 into the equation, resulting in ( x − h ) 2 + y 2 = 36 .
The equations that fit the conditions are: ( x − 6 ) 2 + y 2 = 36 and ( 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. The center of the circle lies on the x-axis, meaning the y-coordinate of the center 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 = 36 . We need to check which of the given equations fit this form.
Check each equation Let's analyze each option:
( x − 12 ) 2 + y 2 = 12 : This equation has a center at ( 12 , 0 ) and a radius squared of 12, meaning the radius is 12 = 2 3 . This doesn't match our radius of 6, so it's incorrect.
( x − 6 ) 2 + y 2 = 36 : This equation has a center at ( 6 , 0 ) and a radius squared of 36, meaning the radius is 36 = 6 . This matches our conditions, so it's a possible equation.
x 2 + y 2 = 12 : This equation has a center at ( 0 , 0 ) and a radius squared of 12, meaning the radius is 12 = 2 3 . This doesn't match our radius of 6, so it's incorrect.
x 2 + y 2 = 144 : This equation has a center at ( 0 , 0 ) and a radius squared of 144, meaning the radius is 144 = 12 . This doesn't match our radius of 6, so it's incorrect.
( x + 6 ) 2 + y 2 = 36 : This equation has a center at ( − 6 , 0 ) and a radius squared of 36, meaning the radius is 36 = 6 . This matches our conditions, so it's a possible equation.
( x + 12 ) 2 + y 2 = 144 : This equation has a center at ( − 12 , 0 ) and a radius squared of 144, meaning the radius is 144 = 12 . This doesn't match our radius of 6, so it's incorrect.
State the 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 essential in various real-world applications. For instance, when designing a circular garden, knowing the center and radius helps determine the placement and size of the garden within a landscape. Similarly, in architecture, circular arches and domes rely on the principles of circle equations to ensure structural integrity and aesthetic appeal. The equation of a circle also plays a crucial role in GPS technology, where determining distances from satellites to a receiver involves solving circle equations to pinpoint location.
