The line

is a tangent to the circle with equation

1. Find the two possible values of $k.$

2. The two possible lines found in part (a) touch the circle at $A$ and $B.$ Prove that $AB$ is a diameter of the circle.

It is slightly more convenient to substitute $x=2y+k$ into the equation of the circle. Again, since each line touches the circle in exactly one point, the discriminant of the result must be $0.$

Find the coordinates of $A$ and $B,$ find their midpoint and make sure this is the centre of the circle to show that $AB$ is a diameter. (You could also show that $|AB|$ is two times the radius.)