Analyzing Tangent Lines to a Circle

Consider a circle with a radius of 3 centered at (0,5). Solve the following problems about this circle and its tangent lines.

1. State the equation for this circle and differentiate it.
2. Find two points with a y-value less than 5 such that the slope of the tangent line at one of these points is 1 and is -1 at the other point. Determine where these tangent lines intersect each other and where they intersect the x-axis. Is there anything in particular that you notice?
3. Prove that for each tangent line on this circle, there is exactly one parallel tangent line.
4. When do two tangent lines intersect on the y-axis? State a function for the y-value at which tangent lines intersect when they intersect on the y-axis.