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 radius of the circle is calculated as half of the diameter: r = 2 12 = 6 , so r 2 = 36 .
The general equation of a circle with center ( 0 , k ) on the y -axis is x 2 + ( y − k ) 2 = 36 .
Option 1, x 2 + ( y − 3 ) 2 = 36 , matches the form with center ( 0 , 3 ) . Option 5, x 2 + ( y + 8 ) 2 = 36 , matches the form with center ( 0 , − 8 ) .
The equations that satisfy the conditions are: x 2 + ( y − 3 ) 2 = 36 and x 2 + ( y + 8 ) 2 = 36 .
Explanation
Understanding the Problem The problem asks us to identify two equations of circles that have a diameter of 12 units and a center lying on the y -axis. Let's break down what this means.
Finding the Radius First, a circle's diameter is twice its radius. Since the diameter is 12, the radius r is 2 12 = 6 . Therefore, r 2 = 6 2 = 36 .
Determining the Circle Equation Form Second, a circle's center lying on the y -axis means the x -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. In our case, h = 0 , so the equation simplifies to x 2 + ( y − k ) 2 = r 2 . Since r 2 = 36 , the equation becomes x 2 + ( y − k ) 2 = 36 .
Checking Each Option Now, let's examine each option to see if it fits this form:
Option 1: x 2 + ( y − 3 ) 2 = 36 . This matches 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.
Option 2: x 2 + ( y − 5 ) 2 = 6 . This does not match the form because the right side is 6, not 36. The radius would be 6 , not 6. This is not a valid option.
Option 3: ( x − 4 ) 2 + y 2 = 36 . This does not match the form because the x -coordinate of the center is 4, not 0. The center is ( 4 , 0 ) , which is not on the y -axis. This is not a valid option.
Option 4: ( x + 6 ) 2 + y 2 = 144 . This does not match the form because the x -coordinate of the center is -6, not 0, and the right side is 144, not 36. The center is ( − 6 , 0 ) , which is not on the y -axis, and the radius is 12, not 6. This is not a valid option.
Option 5: x 2 + ( y + 8 ) 2 = 36 . This matches 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 on the y -axis are x 2 + ( y − 3 ) 2 = 36 and x 2 + ( y + 8 ) 2 = 36 .
Examples
Understanding circles and their equations is crucial in many real-world applications. For instance, when designing a circular garden, you need to know the equation to properly map out the layout. Similarly, in architecture, arches and domes often involve circular segments, and knowing their equations helps in structural calculations and design. Even in computer graphics, circles are fundamental for creating various shapes and animations, making the understanding of their properties essential.
