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 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 the center as ( 0 , 10 ) .
We find the radius by taking the square root of the right-hand side: r = 16 = 4 .
The radius of the circle is 4 units and the center is at ( 0 , 10 ) .
Explanation
Analyze the problem and given data 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. 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.
Compare with the standard form Comparing the given equation x 2 + ( y − 10 ) 2 = 16 with the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , we can identify the center and radius.
Identify the center The x-coordinate of the center, h , is 0 because the equation has x 2 which is equivalent to ( x − 0 ) 2 . The y-coordinate of the center, k , is 10 because we have ( y − 10 ) 2 in the equation. Therefore, the center of the circle is at ( 0 , 10 ) .
Calculate the radius The radius r can be found by taking the square root of the right-hand side of the equation, which is 16. So, $r =
16 = 4
The radius of the circle is 4 units.
State the 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 instance, when designing a circular garden, you need to know the center and radius to plan the layout accurately. Similarly, in computer graphics, circles are fundamental shapes, and their equations are used to draw them on the screen. Knowing the center and radius allows you to position and size the circle correctly in your design or application.
