Which equations represent circles that have a diameter of 12 units and a center that lies on the $y$-axis? Select two options.

$x^2+(y-3)^2=36$
$x^2+(y-5)^2=6$
$(x-4)^2+y^2=36$
$(x+6)^2+y^2=144$
$x^2+(y+8)^2=36$

The problem requires identifying circle equations with a diameter of 12 and a center on the y-axis.
The general form of a circle's equation is ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius. Since the center is on the y-axis, h = 0 and r = 6 .
Substitute h = 0 and r 2 = 36 into the general equation, resulting in x 2 + ( y − k ) 2 = 36 .
Check each given equation to see if it matches this form. The two correct equations are x 2 + ( y − 3 ) 2 = 36 and x 2 + ( y + 8 ) 2 = 36 .
x 2 + ( y − 3 ) 2 = 36 , x 2 + ( y + 8 ) 2 = 36 ​

Explanation

Problem Analysis Let's analyze the problem. We are looking for equations of circles with a diameter of 12 and a center on the y-axis. 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. Since the center lies on the y-axis, h = 0 . The diameter is 12, so the radius is r = 12/2 = 6 . Thus, r 2 = 36 . The equation becomes x 2 + ( y − k ) 2 = 36 . We need to find equations in this form.

Checking the options Now, let's examine the given options:

x 2 + ( y − 3 ) 2 = 36 : This equation is in the form x 2 + ( y − k ) 2 = 36 with k = 3 . So, the center is ( 0 , 3 ) and the radius is 6. This is a valid option.

x 2 + ( y − 5 ) 2 = 6 : This equation is in the form x 2 + ( y − k ) 2 = r 2 with k = 5 and r 2 = 6 . The radius is 6 ​ , not 6. So, this is not a valid option.

( x − 4 ) 2 + y 2 = 36 : This equation is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 with h = 4 , k = 0 , and r 2 = 36 . The center is ( 4 , 0 ) , which is not on the y-axis. So, this is not a valid option.

( x + 6 ) 2 + y 2 = 144 : This equation is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 with h = − 6 , k = 0 , and r 2 = 144 . The center is ( − 6 , 0 ) , which is not on the y-axis. Also, the radius is 12, not 6. So, this is not a valid option.

x 2 + ( y + 8 ) 2 = 36 : This equation is in the form x 2 + ( y − k ) 2 = 36 with k = − 8 . So, the center is ( 0 , − 8 ) and the radius is 6. This is a valid option.

Final Answer Therefore, the two equations that represent circles with a diameter of 12 units and a center that lies on the y -axis are x 2 + ( y − 3 ) 2 = 36 and x 2 + ( y + 8 ) 2 = 36 .


Examples
Understanding the equation of a circle is useful in many real-world applications. For example, when designing a circular garden, you need to know the equation to properly plan the layout and ensure it fits within the available space. Similarly, in architecture, circular windows or domes require precise calculations based on the circle's equation 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.

Answered by GinnyAnswer | 2025-07-08