To find which point has a distance of 10 from the point \((-1, 2)\), we can use the distance formula, which is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where \((x_1, y_1)\) is the original point \((-1, 2)\) and \((x_2, y_2)\) is the point we are testing.
We want the distance \(d\) to equal 10:
\[ \sqrt{(x_2 + 1)^2 + (y_2 - 2)^2} = 10 \]
Squaring both sides, we have:
\[ (x_2 + 1)^2 + (y_2 - 2)^2 = 100 \]
Now, let's check each option:
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For the point \((-1, -12)\): \[ (-1 + 1)^2 + (-12 - 2)^2 = 0 + (-14)^2 = 196 \quad (\text{not } 100) \]
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For the point \((-1, 10)\): \[ (-1 + 1)^2 + (10 - 2)^2 = 0 + 8^2 = 64 \quad (\text{not } 100) \]
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For the point \((-11, 2)\): \[ (-11 + 1)^2 + (2 - 2)^2 = (-10)^2 + 0 = 100 \quad (\text{is } 100) \]
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For the point \((10, 2)\): \[ (10 + 1)^2 + (2 - 2)^2 = 11^2 + 0 = 121 \quad (\text{not } 100) \]
The only point that has a distance of 10 from \((-1, 2)\) is \((-11, 2)\).
Thus, the answer is \((-11, 2)\).