Bird Ortega
08/19/2024 · Senior High School
Write the standard form of the equation of the circle described below. Center \( (0,-4) \) passes through the point \( (-1,2) \) The standard form of the equation of the circle is (Type an equation. Simplify your answer.)
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Step-by-step Solution
To write the standard form of the equation of a circle, we use the formula:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given the center of the circle is \((0, -4)\), we have \(h = 0\) and \(k = -4\).
Next, we need to find the radius \(r\). The radius is the distance from the center to any point on the circle. We know the circle passes through the point \((-1, 2)\). We can calculate the radius using the distance formula:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the center \((0, -4)\) as \((x_1, y_1)\) and the point \((-1, 2)\) as \((x_2, y_2)\):
\[
r = \sqrt{((-1) - 0)^2 + (2 - (-4))^2}
\]
\[
= \sqrt{(-1)^2 + (2 + 4)^2}
\]
\[
= \sqrt{1 + 6^2}
\]
\[
= \sqrt{1 + 36}
\]
\[
= \sqrt{37}
\]
Now we have \(r^2 = 37\).
Substituting \(h\), \(k\), and \(r^2\) into the standard form equation:
\[
(x - 0)^2 + (y - (-4))^2 = 37
\]
This simplifies to:
\[
x^2 + (y + 4)^2 = 37
\]
Thus, the standard form of the equation of the circle is:
\[
\boxed{x^2 + (y + 4)^2 = 37}
\]
Quick Answer
The standard form of the equation of the circle is \(x^2 + (y + 4)^2 = 37\).
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