Question 1114745
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You are not completing the square correctly.  The standard form of the equation for the circle should have (x-3)^2 and (y-6)^2....<br>
{{{x^2 + y^2 - 6x - 12y + 41 = 0}}}
{{{x^2-6x+9 + y^2-12y+36 = -41+9+36}}}
{{{(x-3)^2 + (y-6)^2 = 4}}}<br>
The circle has center (3,6) and radius 2.<br>
The distance from the origin to the center of the circle, by the Pythagorean Theorem, is sqrt(45) = 3*sqrt(5); since the radius of the circle is 2, the shortest distance from the origin to a point on the circle is 3*sqrt(5)-2.