Maths - Coordinate geometry

The line with equation mx-y-2=0 touches the circle with equation x^2 + 6x + y^2 - 8y = 4. find the two possible values of m.

Touch implies it only hits the circle once, the line is tangential to the circle

If you remember, this touching once will mean when you plug in and rearrange for a quadratic in X, the discriminant of the function must equal 0 for a repeated, single root.

rearrange your line to get y = mx - 2, now you can sub that into your circle equation, fiddle around and see what happens.