Point of tangency to a circle from a point not on the circle

1. The problem statement, all variables and given/known data

Find the points of tangency to a circle given by x^2+y^2=9 from point (12,9).





2. Relevant equations

dy/dx=-x/y

(what I've been able to come up so far)





3. The attempt at a solution

Taking the derivative I got dy/dx=-x/y



Let the unknown point of tangency be (a,b)



y-b=(-a/b)(x-a)

Simplifying that, I got:

by-ax=a^2+b^2

a and b fall on the circle; the circle's equation is x^2+y^2=9; therefore, a^2+b^2=9



by-ax=9

(12,9) is a point on this ^ line, so



9b-12a=9

b=(4/3)a+1



Substituting back into the original equation x^2+y^2=9,



a^2+((4/3)a+1)^2=9



Simplifying that got me 25a^2+27a-72=0.



This was the point where I knew I was wrong. Where did I go wrong/how do I fix it?