Alevel maths equations of lines

Try making y the subject in the equation of the line...

I tried that and I ended up with x^2+6x+m^2x^2-12mx+16=0 and I didn't know where to go after that because there's two unkowns

You want the value(s) of m where there is a single solution to the quadratic equation in x. If there are no solutions, the line does not cut the circle. If there are two solutions, the line cuts the circle. If there is one solution, the line is a tangent to the circle.

So write the quadratic in terms of x, then use the quadratic formula and find the values of m where the discriminant is zero.

Sorry but I don't really understand what you mean by that

Just sketch the differerent scenarios. The line could
* miss the circle,
* graze the circumference (tangent)
* cut the circle and intersect the circumference in two places

For the quadratic in x you have, this corresponds to 0, 1 and 2 solutions. A quadratic has 1 solution (repeated) when the discriminant is zero. So write down the discriminant and set it equal to zero to find the values of m which give lines which are tangent to the circle.

I tried that and I ended up with x^2+6x+m^2x^2-12mx+16=0 and I didn't know where to go after that because there's two unkowns

You may wish to group like terms..

It's the part about putting x^2+6x+m^2x^2-12mx+16=0 into a quadratic that's Confusing me because I dont see how you would do it

(some of the) coefficients depend on m.
x^2*() + x*() + () =