the line y=ax + k is tangent to the circle xsquared + (y-4)squared at the point (4,6). What is the value of (a+k)?
I would solve this problem by first understanding that the line tangent to the circle will be perpindicular to a line going through the given point, and the ceter of the circle, which can be found easily. Once you find the slope of the line through the center, all you have to do is insert the negative reciprocal into the eequation where you see (a), solve for k and the rest is history!
Draw your circle. Now draw a line from the origin to the point (4,6) on that circle. The line tangent to this point will be perpendicular to line you just drew. As such, the slope of the perpendicular must be -1/(m) for the slope of the drawn line.
Like, if the line you drew had a slope of 2, the tangent would have a slope of -1/2.