We first get

x²+y²-10x+9 = 0

in standard form.

(x-5)²+(y-0)²=16

The center is (5,0) and the point (3,1) is the midpoint
of the chord.



Draw in the two green radii and the red line from the 
center to the midpoint of the chord (3,1).



The two right triangles are congruent by SSS, so it's easy to
prove that the red line is perpendicular to the chord.

So find the slope of the red line.
Then the slope of the chord will be the negative reciprocal
of the slope of the red line.

Then since the chord passes through (3,1), you will use
the point-slope formula to get the equation of the chord.

If you have trouble, let me know in the thank-you note form 
below and I will get back to you by email.

Edwin