Question
Which equation is the standard form of the equation of a circle drawn on the coordinate plane, derived from the Pythagorean Theorem? (1 point) Responses (x−2)2+(y−2)2=4 left parenthesis x minus 2 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 4 (x−2)2+(y+2)2=16 left parenthesis x minus 2 right parenthesis squared plus left parenthesis y plus 2 right parenthesis squared equals 16 (x−6)2+(y−2)2=16 left parenthesis x minus 6 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 16 (x+2)2+(y−6)2=4
Answers
GPT-4o mini
The standard form of the equation of a circle is given by the formula:
$$(x - h)^2 + (y - k)^2 = r^2$$
where \((h, k)\) is the center of the circle, and \(r\) is the radius.
Looking at the options provided, let's analyze each one:
1. \((x - 2)^2 + (y - 2)^2 = 4\)
- Center: (2, 2)
- Radius: \(r = \sqrt{4} = 2\)
- This is in standard form.
2. \((x - 2)^2 + (y + 2)^2 = 16\)
- Center: (2, -2)
- Radius: \(r = \sqrt{16} = 4\)
- This is also in standard form.
3. \((x - 6)^2 + (y - 2)^2 = 16\)
- Center: (6, 2)
- Radius: \(r = \sqrt{16} = 4\)
- This is in standard form as well.
4. \((x + 2)^2 + (y - 6)^2 = 4\)
- Center: (-2, 6)
- Radius: \(r = \sqrt{4} = 2\)
- This is also in standard form.
Thus, all the provided equations are valid equations of circles written in standard form derived from the Pythagorean Theorem.
If you are looking for just one correct answer as per usual conventions, the first option, \((x - 2)^2 + (y - 2)^2 = 4\), is often the simplest and most recognizable form, as it has a radius of 2 and a center that is easy to identify.
So, the correct response is:
\((x - 2)^2 + (y - 2)^2 = 4\).
$$(x - h)^2 + (y - k)^2 = r^2$$
where \((h, k)\) is the center of the circle, and \(r\) is the radius.
Looking at the options provided, let's analyze each one:
1. \((x - 2)^2 + (y - 2)^2 = 4\)
- Center: (2, 2)
- Radius: \(r = \sqrt{4} = 2\)
- This is in standard form.
2. \((x - 2)^2 + (y + 2)^2 = 16\)
- Center: (2, -2)
- Radius: \(r = \sqrt{16} = 4\)
- This is also in standard form.
3. \((x - 6)^2 + (y - 2)^2 = 16\)
- Center: (6, 2)
- Radius: \(r = \sqrt{16} = 4\)
- This is in standard form as well.
4. \((x + 2)^2 + (y - 6)^2 = 4\)
- Center: (-2, 6)
- Radius: \(r = \sqrt{4} = 2\)
- This is also in standard form.
Thus, all the provided equations are valid equations of circles written in standard form derived from the Pythagorean Theorem.
If you are looking for just one correct answer as per usual conventions, the first option, \((x - 2)^2 + (y - 2)^2 = 4\), is often the simplest and most recognizable form, as it has a radius of 2 and a center that is easy to identify.
So, the correct response is:
\((x - 2)^2 + (y - 2)^2 = 4\).