The equation of a circle is $x^2+(y-10)^2=16$.
The radius of the circle is $\square$ units.
The center of the circle is at $\square$
Answer (1)
The problem provides the equation of a circle: x 2 + ( y − 10 ) 2 = 16 .
We identify the equation's center by comparing it to the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , finding the center at ( 0 , 10 ) .
We determine the radius by recognizing that r 2 = 16 , so r = 16 = 4 .
The radius of the circle is 4 units and the center is at ( 0 , 10 ) .
Explanation
Problem Analysis The equation of a circle is given as x 2 + ( y − 10 ) 2 = 16 . We need to find the radius and the center of the circle.
Standard Form of Circle Equation The standard 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.
Identifying the Center Comparing the given equation x 2 + ( y − 10 ) 2 = 16 with the standard form, we can identify the center and radius.
x 2 can be written as ( x − 0 ) 2 . So, h = 0 .
( y − 10 ) 2 implies k = 10 .
Therefore, the center of the circle is ( 0 , 10 ) .
Calculating the Radius The right-hand side of the equation is 16, which corresponds to r 2 . Therefore, r 2 = 16 .
To find the radius r , we take the square root of 16: r = 16 = 4 .
Final Answer The radius of the circle is 4 units, and the center of the circle is at ( 0 , 10 ) .
Examples
Understanding the equation of a circle is crucial in various real-world applications. For instance, when designing a circular garden, the equation helps determine the layout and boundaries. If you want a circular garden with a radius of 4 meters centered 10 meters away from a reference point, you can use the equation ( x − 0 ) 2 + ( y − 10 ) 2 = 4 2 to define its precise location and size. This ensures accurate planning and efficient use of space.
