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 compare this equation to the general form of a circle's equation: ( x − a ) 2 + ( y − b ) 2 = r 2 , where ( a , b ) is the center and r is the radius.
By comparing the given equation with the general form, we identify the center as ( 0 , 10 ) and r 2 = 16 .
Taking the square root of 16, we find the radius r = 4 . Therefore, 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.
General Equation of a Circle The general equation of a circle is ( x − a ) 2 + ( y − b ) 2 = r 2 , where ( a , b ) is the center and r is the radius.
Rewrite the Given Equation Comparing the given equation x 2 + ( y − 10 ) 2 = 16 with the general equation, we can rewrite the given equation as ( x − 0 ) 2 + ( y − 10 ) 2 = 4 2 .
Identify Center and Radius From the rewritten equation, we can identify the center of the circle as ( 0 , 10 ) and the radius as r = 4 .
Final Answer Therefore, 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, knowing the center and radius helps in accurately planning the layout and determining the amount of fencing needed. Similarly, in architecture, circular arches and domes rely on precise calculations of the circle's parameters to ensure structural integrity and aesthetic appeal. The equation of a circle also finds applications in fields like GPS navigation, where distances from satellites (approximated as spheres) to a receiver are used to determine location.
