Which equations represent circles that have a diameter of 12 units and a center that lies on the $y$-axis? Select two options.
$x^2+(y-3)^2=36$
$x^2+(y-5)^2=6$
$(x-4)^2+y^2=36$
$(x+6)^2+y^2=144$
$x^2+(y+8)^2=36$
Answer (1)
The 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 y-axis, h = 0 , and the diameter is 12, the radius is r = 6 , so r 2 = 36 .
The equation of the circle is in the form x 2 + ( y − k ) 2 = 36 .
The two equations that satisfy these conditions 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 , and r 2 = 36 . Therefore, we are looking for equations of the form x 2 + ( y − k ) 2 = 36 .
Checking the options Now, let's examine each option:
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 2 = 36 , with h = 4 . 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 2 = r 2 , with h = − 6 and r 2 = 144 . The center is ( − 6 , 0 ) , which is not on the y-axis, and 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 crucial in many real-world applications. For instance, when designing a circular garden, you need to determine the placement and size of the sprinkler system to ensure complete coverage. The center of the circle represents the location of the sprinkler, and the radius determines the area it can irrigate. By knowing the equation of the circle, you can accurately plan the layout and optimize the water distribution for your garden, ensuring every plant receives the necessary hydration.
