Put the line in point-slope form:
y = (7/24)x +(625/24)
The segment that is the radius from the center of the circle to the point of tangency will intersect this line at right angles at the point of tangency.
So this radial segment is along the line: (the slope is the negative reciprocal so that it is perpendicular, at right angles)
y = (-24/7)x which also goes through the circle center at (0,0) as desired.
To find the point of tangency we find where these two lines intersect, setting the y's equal:
(7/24)x + (625/24) = (-24/7)x
Multiply all terms by 7*24 to get rid of the fractions:
7*7*x + 625*7 = -24*24*x
49x + 4375 = -576x
4375 = -625x
x = -7 (wow, it's a nice number!)
and then the y of tangency is:
y = (-24/7)*(-7) = 24
So the point of tangency is (-7,24)
(Sorry that I am not able to include a diagram...)