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 problem gives a circle with a diameter of 12 and a center on the x-axis.
The radius is calculated as half the diameter: r = 2 12 = 6 , so r 2 = 36 .
The general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , and since the center is on the x-axis, k = 0 .
By substituting the values, the possible equations are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 .
The final answer is: ( x − 6 ) 2 + y 2 = 36 , ( 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 determine which of the given equations could represent this circle.
General Equation of a Circle The general equation of a circle with center ( h , k ) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2
Calculate the Radius Since the diameter is 12 units, the radius r is half of the diameter, so r = 2 12 = 6 . Therefore, r 2 = 6 2 = 36 .
Apply the Center Condition Since the center of the circle lies on the x-axis, the y-coordinate of the center is 0. Thus, k = 0 . The equation of the circle becomes: ( x − h ) 2 + y 2 = 36
Check Each Equation Now, we check each of the given equations to see if they fit this form:
( x − 12 ) 2 + y 2 = 12 : This equation has r 2 = 12 , which is not equal to 36. So, it's incorrect.
( x − 6 ) 2 + y 2 = 36 : This equation has h = 6 and r 2 = 36 . So, it's correct.
x 2 + y 2 = 12 : This equation has h = 0 and r 2 = 12 , which is not equal to 36. So, it's incorrect.
x 2 + y 2 = 144 : This equation has h = 0 and r 2 = 144 , which is not equal to 36. So, it's incorrect.
( x + 6 ) 2 + y 2 = 36 : This equation has h = − 6 and r 2 = 36 . So, it's correct.
( x + 12 ) 2 + y 2 = 144 : This equation has h = − 12 and r 2 = 144 , which is not equal to 36. So, it's incorrect.
Final Answer Therefore, the correct equations 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 example, when designing a circular garden, you need to know the equation to accurately map out the boundaries. If you want the center of the garden to be 6 units to the right of your starting point and the garden to have a radius of 6 units, the equation ( x − 6 ) 2 + y 2 = 36 will help you define the garden's perimeter precisely. This ensures that your garden design fits perfectly within the intended space, combining mathematical accuracy with practical landscaping.
The valid equations for the circle with a diameter of 12 units and a center on the x-axis are ( x − 6 ) 2 + y 2 = 36 and ( x + 6 ) 2 + y 2 = 36 .
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