To solve for the radius and the center of the circle given the equation [tex]\( x^2 + (y - 10)^2 = 16 \)[/tex], follow these steps:
1. Identify the standard form of a circle's equation:
The standard form of a circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\( (h, k) \)[/tex] is the center and [tex]\( r \)[/tex] is the radius.
2. Compare the given equation to the standard form:
The given equation is [tex]\( x^2 + (y - 10)^2 = 16 \)[/tex].
3. Identify the center:
- By comparing [tex]\( x^2 \)[/tex] with [tex]\( (x - h)^2 \)[/tex], we see that [tex]\( h = 0 \)[/tex].
- By comparing [tex]\( (y - 10)^2 \)[/tex] with [tex]\( (y - k)^2 \)[/tex], we see that [tex]\( k = 10 \)[/tex].
Therefore, the center of the circle is at [tex]\( (0, 10) \)[/tex].
4. Determine the radius:
- The right side of the equation is [tex]\( 16 \)[/tex], which in the standard form represents [tex]\( r^2 \)[/tex].
- Solving for [tex]\( r \)[/tex], we take the square root of both sides: [tex]\( r = \sqrt{16} = 4 \)[/tex].
Hence, the radius of the circle is [tex]\( 4 \)[/tex] units and the center of the circle is at [tex]\( (0, 10) \)[/tex].
