What is the equation of the line tangent line to the circle with center (3,4) at the point (6,0)?

The tangent line is the line the touches the circle and one point, in this case the line will touch at (6,0).  If you draw a segment from the center (3,4) to the point (6,0) the tangent line will be perpendicular to that segment. Perpendicular lines have slopes that are negative reciprocals of each other, so the slopes will multiply to equal -1.

So the slope of the segment from (3,4) to (6,0) is (0-4)/(6-3) = -4/3.  Therefore the slope of the line perpendicular is 3/4. 

3/4 times -4/3 = -1, which is what the slopes of perpendicular lines must multiply to.

Now take the equation y = mx + b where m is the slope and b is the y-intercept. Remember the y-intercept is where the line crosses the y-axis.

We know m = 3/4, we know x = 6 and y = 0.

So put the numbers in and solve for b.

0 = (3/4)(6) + b

0 = 4.5 + b

b = -4.5 or - 9/2

So the equation of the line is y = (3/4)x - (9/2)