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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−2)2+(y+3)2=3open paren x minus 2 close paren squared plus open paren y plus 3 close paren squared is equal to 3

(x−2)2+(y+3)2=9open paren x minus 2 close paren squared plus open paren y plus 3 close paren squared is equal to 9

(x+2)2+(y−3)2=3open paren x plus 2 close paren squared plus open paren y minus 3 close paren squared is equal to 3

(x+2)2+(y−3)2=9open paren x plus 2 close paren squared plus open paren y minus 3 close paren squared is equal to 9

1 answer

The standard form of the equation of a circle 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.

To determine which of the provided equations is in the standard form, we need to identify both the center and the square of the radius for each option:

  1. \( (x−2)^2 + (y+3)^2 = 3 \)

    • Center: \((2, -3)\)
    • Radius: \(\sqrt{3}\)

  2. \( (x−2)^2 + (y+3)^2 = 9 \)

    • Center: \((2, -3)\)
    • Radius: \(3\)

  3. \( (x+2)^2 + (y−3)^2 = 3 \)

    • Center: \((-2, 3)\)
    • Radius: \(\sqrt{3}\)

  4. \( (x+2)^2 + (y−3)^2 = 9 \)

    • Center: \((-2, 3)\)
    • Radius: \(3\)

All four equations are in the standard form of the equation of a circle. However, if we're looking for a specific one, we should consider the radii \(r^2\).

If you're choosing based on the radius, the equations with \(r^2 = 9\) would provide a larger circle than those with \(r^2 = 3\).

Therefore, the equations with \(9\) as the radius squared are \( (x−2)^2 + (y+3)^2 = 9 \) and \( (x+2)^2 + (y−3)^2 = 9 \). Any of these can be considered valid; please specify if you need a specific choice based on the radius or center!