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
x2+y2=25
x squared plus y squared equals 25
(x+4)2+(y+3)2=25
left parenthesis x plus 4 right parenthesis squared plus left parenthesis y plus 3 right parenthesis squared equals 25
(x−4)2+(y−3)2=25
left parenthesis x minus 4 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25
x2+(y−5)2=25
(1 point)
Responses
x2+y2=25
x squared plus y squared equals 25
(x+4)2+(y+3)2=25
left parenthesis x plus 4 right parenthesis squared plus left parenthesis y plus 3 right parenthesis squared equals 25
(x−4)2+(y−3)2=25
left parenthesis x minus 4 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 25
x2+(y−5)2=25
Answers
GPT-4o mini
The standard form of the equation of a circle in the coordinate plane 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.
Among the provided options, the equations that fit this format are:
1. \((x + 4)^2 + (y + 3)^2 = 25\), which has a center at \((-4, -3)\) and a radius of \(5\).
2. \((x - 4)^2 + (y - 3)^2 = 25\), which has a center at \((4, 3)\) and a radius of \(5\).
3. \(x^2 + (y - 5)^2 = 25\), which is equivalent to \((x - 0)^2 + (y - 5)^2 = 25\), making it a circle centered at \((0, 5)\) with a radius of \(5\).
The first two options are more explicit in showing the center, while the last option can be interpreted as a circle whose center is at the origin in the x-direction.
Thus, all three, except the first option, meet the standard form of the equation of a circle based on the structure provided.
If you are looking for the equation that visually indicates a traditional circle equation most commonly derived from Pythagorean theorem, the two forms with transformations are the standard form; hence the most suitable answers would be:
1. \((x + 4)^2 + (y + 3)^2 = 25\)
2. \((x - 4)^2 + (y - 3)^2 = 25\)
Choose either of those as they are standard forms of a circle's equation.
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Among the provided options, the equations that fit this format are:
1. \((x + 4)^2 + (y + 3)^2 = 25\), which has a center at \((-4, -3)\) and a radius of \(5\).
2. \((x - 4)^2 + (y - 3)^2 = 25\), which has a center at \((4, 3)\) and a radius of \(5\).
3. \(x^2 + (y - 5)^2 = 25\), which is equivalent to \((x - 0)^2 + (y - 5)^2 = 25\), making it a circle centered at \((0, 5)\) with a radius of \(5\).
The first two options are more explicit in showing the center, while the last option can be interpreted as a circle whose center is at the origin in the x-direction.
Thus, all three, except the first option, meet the standard form of the equation of a circle based on the structure provided.
If you are looking for the equation that visually indicates a traditional circle equation most commonly derived from Pythagorean theorem, the two forms with transformations are the standard form; hence the most suitable answers would be:
1. \((x + 4)^2 + (y + 3)^2 = 25\)
2. \((x - 4)^2 + (y - 3)^2 = 25\)
Choose either of those as they are standard forms of a circle's equation.