To find the center and radius of the circle defined by the equation \( x^2 + (y + 4)^2 = 3 \), we can compare it to the standard form of a circle's equation, which is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Looking at the equation \( x^2 + (y + 4)^2 = 3 \):
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Identifying the center:
- The expression \( (y + 4)^2 \) can be rewritten as \( (y - (-4))^2 \). Thus, we can determine \(k = -4\) and \(h = 0\).
- So, the center is \((0, -4)\).
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Identifying the radius:
- The right side of the equation is \(3\), which corresponds to \(r^2\). Therefore, \(r^2 = 3\).
- To find \(r\), we take the square root: \[ r = \sqrt{3} \]
Thus, the center of the circle is \((0, -4)\) and the radius is \(\sqrt{3}\).
Based on the options provided, the correct answer is:
center: (0, -4); r = √3