What are the possible values of k in the line y equals 2x plus k that makes contact with the circle x2 plus y2 equals 4?

If: y = 2x+k then y^2 = 4x^2 +4kx +k^2

If: x^2 +y^2 = 4 then 5x^2 +4kx +(k^2 -4) = 0

Using the discriminant: (4k)^2 -4*5*(k^2 -4) = 0

Removing brackets: 16k^2 -20k^2 +80 = 0

Collecting like terms and subtracting 80 from both sides: -4k^2 = -80

Dividing both sides by -4: k^2 = 20

Square root both sides: k = - square root of 20 and k = + square root of 20

Therefore possible values of k are - square root of 20 and + square root of 20