We have a line with the equation y=(-4/3)x+4.
The circle has a radius 1, and the top of the circle also has the tangent line y=2, and the bottom has a tangent line y=0.
We need to find k in the diagram below or at least the point where y=(-4/3)x+4 intersects the circle.
We are at a loss on how to solve this, but are sure that we're overthinking this. Can anyone help us get started?
EDIT: the center of the circle should be (k, 1), not (k, 2).
Just to verify--is the y-coordinate of your circle at 2 or at 1?
The equation for the circle is
(x-k)^2 + (y-1)^2 = 1.
The equation for the line is
y = (-4/3)x + 4.
Substitute this expression for y into the equation for the circle to get a new equation in terms of only x and k. Expand the terms to get a quadratic in x. Use the quadratic formula to solve for x in terms of k.
Depending on the value of k, your circle will either intersect the line at exactly two points, exactly one point, or at no points. Try to determine which values of k will result in only one intersection.
Hope this helps.