Hills Horton
01/20/2023 · Senior High School
Find an equation of the circle that has center \( (-4,4) \) and passes through \( (2,-6) \).
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the equation of a circle, we use the standard form of the equation of a circle, which is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given the center of the circle \((-4, 4)\), we have:
\[
h = -4 \quad \text{and} \quad k = 4
\]
Next, we need to find the radius \(r\). The radius is the distance from the center of the circle to any point on the circle. We are given that the circle passes through the point \((2, -6)\). We can calculate the distance between the center \((-4, 4)\) and the point \((2, -6)\) using the distance formula:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of the center and the point:
\[
r = \sqrt{(2 - (-4))^2 + (-6 - 4)^2}
\]
Calculating the differences:
\[
r = \sqrt{(2 + 4)^2 + (-6 - 4)^2} = \sqrt{(6)^2 + (-10)^2}
\]
Calculating the squares:
\[
r = \sqrt{36 + 100} = \sqrt{136}
\]
Simplifying \(\sqrt{136}\):
\[
r = \sqrt{4 \cdot 34} = 2\sqrt{34}
\]
Now that we have the radius, we can substitute \(h\), \(k\), and \(r\) into the equation of the circle:
\[
(x - (-4))^2 + (y - 4)^2 = (2\sqrt{34})^2
\]
This simplifies to:
\[
(x + 4)^2 + (y - 4)^2 = 4 \cdot 34
\]
Calculating \(4 \cdot 34\):
\[
4 \cdot 34 = 136
\]
Thus, the equation of the circle is:
\[
(x + 4)^2 + (y - 4)^2 = 136
\]
This is the final equation of the circle.
Quick Answer
The equation of the circle is \((x + 4)^2 + (y - 4)^2 = 136\).
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