You were actually completely along the right lines! The only thing you did wrong was not trusting your instincts and having confidence in the method.
Here's the full explanation:
or more simply
As line is a tangent to the circle , we can work out the point they intersect by either using a simultaneous equation, or more simply, as you did, substituting in terms of into the equation for .
now by tidying up,
all coefficients have a factor of 17, so we can simplify further:
n.b - I did not know off the top of my head that all those numbers are divisible by 17, but you can assume that it'll be something simple like that and then test to see if you are right (by quickly using your calculator). The questions in the books and exams are usually nice and tidy like that. So don't be put off when you see some quadratic with huge numbers - it's just trying to put you off!
Now this equation looks much more simple. We can easily solve to find :
We've got a repeated root which is great because we are told the line and circle on intersect at one point (the tangent).
substitution the value for into the equation for line :
So the coordinates of are .