Question
A circle is drawn on a coordinate plane with the center point at (-4, 6) and a radius of 2. Derive the equation of the circle from the given information. Fill in the missing information in the following equation. (2 points)
(X- box )2+ (y- box)2 = box
(X- box )2+ (y- box)2 = box
Answers
Answered by
GPT-4o mini
To derive the equation of the circle given the center \((-4, 6)\) and radius \(2\), we use the standard form of the equation of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
In this case:
- The center \((h, k)\) is \((-4, 6)\), so \(h = -4\) and \(k = 6\).
- The radius \(r\) is \(2\), so \(r^2 = 2^2 = 4\).
Now, we can substitute these values into the standard form:
\[
(x - (-4))^2 + (y - 6)^2 = 4
\]
This simplifies to:
\[
(x + 4)^2 + (y - 6)^2 = 4
\]
Now we can fill in the missing information in the equation format you provided:
\[
(X - \text{-4})^2 + (y - 6)^2 = 4
\]
So the filled boxes are:
1. box 1: \(-4\)
2. box 2: \(6\)
3. box 3: \(4\)
Thus, the final equation is:
\[
(X + 4)^2 + (y - 6)^2 = 4
\]
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
In this case:
- The center \((h, k)\) is \((-4, 6)\), so \(h = -4\) and \(k = 6\).
- The radius \(r\) is \(2\), so \(r^2 = 2^2 = 4\).
Now, we can substitute these values into the standard form:
\[
(x - (-4))^2 + (y - 6)^2 = 4
\]
This simplifies to:
\[
(x + 4)^2 + (y - 6)^2 = 4
\]
Now we can fill in the missing information in the equation format you provided:
\[
(X - \text{-4})^2 + (y - 6)^2 = 4
\]
So the filled boxes are:
1. box 1: \(-4\)
2. box 2: \(6\)
3. box 3: \(4\)
Thus, the final equation is:
\[
(X + 4)^2 + (y - 6)^2 = 4
\]