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 (2)
The equation of the circle is in the form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius.
By comparing the given equation x 2 + ( y − 10 ) 2 = 16 with the standard form, we identify h = 0 , k = 10 , and r 2 = 16 .
We calculate the radius by taking the square root of 16, which gives r = 4 .
The center of the circle is ( 0 , 10 ) and the radius is 4 units, so the final answer is 4 and ( 0 , 10 ) .
Explanation
Analyze the problem and given data The equation of a circle is given by x 2 + ( y − 10 ) 2 = 16 . We need to find the radius and the center of this circle. The general equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . Comparing the given equation with the general equation, we can identify the center and the radius.
Identify the center and radius Comparing x 2 + ( y − 10 ) 2 = 16 with ( x − h ) 2 + ( y − k ) 2 = r 2 , we can see that:
x 2 can be written as ( x − 0 ) 2 , so h = 0 .
( y − 10 ) 2 implies k = 10 .
r 2 = 16 , so r = 16 .
Calculate the radius To find the radius, we take the square root of 16: r = 16 = 4 So, the radius of the circle is 4 units.
State the center The center of the circle is at ( h , k ) = ( 0 , 10 ) .
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 useful in various real-world applications. For example, when designing a circular garden, you need to know the center and radius to plan the layout accurately. Similarly, in computer graphics, circles are frequently used, and knowing their equation helps in drawing them on the screen. Also, in navigation, the range of a radar or GPS can be modeled as a circle, where the center is the location of the device and the radius is the maximum range.
The center of the circle is at (0, 10) and the radius is 4 units.
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