If 3x+y-5=0 is the equation of a chord of the circle x^{2}+y^{2}-25=0, then what are the coordinates of the mid-point of the chord?

Explanation

The line from the center (0,0) to the chord's mid-point is perpendicular to the chord. The slope of the chord 3x+y-5=0 is -3. The perpendicular line from the origin thus has a slope of \frac{1}{3}, giving its equation as y = \frac{1}{3}x, or x=3y. Substituting this into the chord's equation yields 3(3y)+y=5 \implies 10y=5 \implies y=\frac{1}{2}. Thus x = \frac{3}{2}. The mid-point is (\frac{3}{2}, \frac{1}{2}).

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