Question
Determine the equation of the circle with center left parenthesis, minus, 7, comma, 7, right parenthesis(−7,7) containing the point left parenthesis, minus, 2, comma, minus, 5, right parenthesis(−2,−5).
Answers
GPT-4o mini
To determine the equation of the circle, we start from the standard form of the circle's equation, 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.
In this problem, the center of the circle is given as \((-7, 7)\). Therefore, we have:
\[
h = -7 \quad \text{and} \quad k = 7
\]
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. Here, we will use the point \((-2, -5)\) which lies on the circle to calculate the radius.
We can use the distance formula to find the distance \(r\):
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting in our points, where \((x_1, y_1) = (-7, 7)\) and \((x_2, y_2) = (-2, -5)\):
\[
r = \sqrt{((-2) - (-7))^2 + ((-5) - 7)^2}
\]
Calculating each component,
\[
(-2) - (-7) = -2 + 7 = 5
\]
\[
(-5) - 7 = -5 - 7 = -12
\]
Now, substituting these values back into the distance formula:
\[
r = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Thus, the radius \(r\) is 13.
Now we can substitute \(h\), \(k\), and \(r\) back into the standard form of the circle's equation:
\[
(x - (-7))^2 + (y - 7)^2 = 13^2
\]
This simplifies to:
\[
(x + 7)^2 + (y - 7)^2 = 169
\]
This is the equation of the circle. Therefore, the final answer is:
\[
\boxed{(x + 7)^2 + (y - 7)^2 = 169}
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
In this problem, the center of the circle is given as \((-7, 7)\). Therefore, we have:
\[
h = -7 \quad \text{and} \quad k = 7
\]
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. Here, we will use the point \((-2, -5)\) which lies on the circle to calculate the radius.
We can use the distance formula to find the distance \(r\):
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting in our points, where \((x_1, y_1) = (-7, 7)\) and \((x_2, y_2) = (-2, -5)\):
\[
r = \sqrt{((-2) - (-7))^2 + ((-5) - 7)^2}
\]
Calculating each component,
\[
(-2) - (-7) = -2 + 7 = 5
\]
\[
(-5) - 7 = -5 - 7 = -12
\]
Now, substituting these values back into the distance formula:
\[
r = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]
Thus, the radius \(r\) is 13.
Now we can substitute \(h\), \(k\), and \(r\) back into the standard form of the circle's equation:
\[
(x - (-7))^2 + (y - 7)^2 = 13^2
\]
This simplifies to:
\[
(x + 7)^2 + (y - 7)^2 = 169
\]
This is the equation of the circle. Therefore, the final answer is:
\[
\boxed{(x + 7)^2 + (y - 7)^2 = 169}
\]