Bot GPT 4-o mini To find the radius of the circle given the equation \(x^2 + y^2 - 14x - 4y = 11\), we will complete the square for both the \(x\) and \(y\) terms.
Step 1: Rearranging the equation Start with the given equation: \[ x^2 + y^2 - 14x - 4y = 11 \] We can rearrange it to group the \(x\) terms and the \(y\) terms together: \[ x^2 - 14x + y^2 - 4y = 11 \]
Step 2: Completing the square for \(x\) To complete the square for the \(x\) terms \(x^2 - 14x\):
- Take half of the coefficient of \(x\) (which is \(-14\)), square it: \((-14/2)^2 = 49\).
- Add and subtract \(49\) inside the equation: \[ (x^2 - 14x + 49) - 49 \] This becomes: \[ (x - 7)^2 - 49 \]
Step 3: Completing the square for \(y\) Next, complete the square for the \(y\) terms \(y^2 - 4y\):
- Take half of the coefficient of \(y\) (which is \(-4\)), square it: \((-4/2)^2 = 4\).
- Add and subtract \(4\): \[ (y^2 - 4y + 4) - 4 \] This becomes: \[ (y - 2)^2 - 4 \]
Step 4: Substitute back into the equation Now substitute back these completed squares into the equation: \[ ((x - 7)^2 - 49) + ((y - 2)^2 - 4) = 11 \] This simplifies to: \[ (x - 7)^2 + (y - 2)^2 - 53 = 11 \] Adding \(53\) to both sides gives: \[ (x - 7)^2 + (y - 2)^2 = 64 \]
Step 5: Identify the center and radius Now, we can identify the standard form of the circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
From our equation:
- The center \((h, k) = (7, 2)\)
- The right side \(64\) means \(r^2 = 64\)
Thus, the radius \(r\) is: \[ r = \sqrt{64} = 8 \]
Therefore, the radius of the circle is \( \boxed{8} \).
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