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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−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−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−6)2=4
left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 6 right parenthesis squared equals 4

(x−2)2+(y+2)2=16

1 answer

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.

From the provided options, each equation is already in this format. Let's analyze each option:

  1. \((x−6)^2+(y−2)^2=16\)

    • Center: \((6, 2)\), Radius: \(\sqrt{16} = 4\)

  2. \((x−2)^2+(y−2)^2=4\)

    • Center: \((2, 2)\), Radius: \(\sqrt{4} = 2\)

  3. \((x+2)^2+(y−6)^2=4\)

    • Center: \((-2, 6)\), Radius: \(\sqrt{4} = 2\)

  4. \((x−2)^2+(y+2)^2=16\)

    • Center: \((2, -2)\), Radius: \(\sqrt{16} = 4\)

All these equations are valid representations of circles. If the question asks which one specifically relates to the Pythagorean theorem's principle of distances on the coordinate plane, any of these equations could apply, as they represent circles based on the theorem.

However, there isn't a "standard" choice among them without further context or criteria provided.

If you need to select one, it might depend on the specific attributes (center or radius) you are looking for, but there’s no incorrect option here in terms of being a valid circle equation.