Question

A circle is drawn on a coordinate plane with the center point at (4, −5) and a radius of 7. Derive the equation of the circle in standard form, using the given information and the Pythagorean theorem. (1 point) Responses (x−4)2+(y+5)2=49 open paren x minus 4 close paren squared plus open paren y plus 5 close paren squared is equal to 49 (x−4)2+(y+5)2=7 open paren x minus 4 close paren squared plus open paren y plus 5 close paren squared is equal to 7 (x+4)2+(y−5)2=49 open paren x plus 4 close paren squared plus open paren y minus 5 close paren squared is equal to 49 (x+4)2+(y+5)2=49

Answer 1

To derive the equation of a circle in standard form, we use the formula:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

In this case, the center of the circle is \((4, -5)\) and the radius \(r\) is 7.

1. Substitute \(h = 4\), \(k = -5\), and \(r = 7\) into the formula:

   • \(h = 4\)
   • \(k = -5\)
   • \(r^2 = 7^2 = 49\)

2. This gives us:

\[ (x - 4)^2 + (y + 5)^2 = 49 \]

Thus, the equation of the circle in standard form is:

\[ (x - 4)^2 + (y + 5)^2 = 49 \]

Among the provided responses, the correct one is:

\[ (x−4)^{2}+(y+5)^{2}=49 \]