1. All lines passing through Q have the equation: y = m (x -q)
2. Determine the Points of intersection between the line and the ellipse:
x^2/a^2 + (m(x-q))^2/b^2 = 1
DSolve for x.
3. To get a tangent point the discriminant must be zero. And this is the additional condition to find the tangent line.