A line tangent to an implicit function, and through two points not on the function

I have been given the implicit function: X^2+4y^2=36, and have been asked to find a line tangent to it through the point (12,3), which is not on the curve.

I know there is one horizontal tangent line, at y=3. However, I am having trouble finding the non=horizontal line.

I know that the derivative is -x/4y. That is fairly simple. I think that I have to set the equation of the line through (12,3) equal to the implicit function, and solve for the slope (which is the derivative) in order to get a numerical value. But I am lost as to how to do that exactly.

(-x/4y)(x-120+3=x^2+4y^2-36

Am I on the right track with this?

Thank you for any help

Respectfully

Physics Student