Soroban's solution is perfect.
While your idea is correct, some mistakes are found in your description.
The tangent of a point on a line is the derivative of the line.
This is incorrect. It is the gradient of the tangent
at a point on the curve that is given by the derivative of y=x^2.
Using the two points (1,1) and (-2,4) we find the two tangent lines are y=2x and y=-4x.
At the point (1,1), the gradient of the tangent =2(1)=2,
hence the equation of the tangent is given by
y-1 = 2(x-1)
Similarly, at (-2,4), the equation of the tangent can be shown to be