I shared my posts (here and here) on the Invariable Calculus Project in the AP Calculus Community. Gary Litvin posted a response within the Community offering there a great non-calculus alternative solution to the original problem of the area of the triangle formed by the x- and y-axes and any Quadrant I tangent line to . Here’s a paraphrase of Gary’s approach.
Let be any Quadrant I tangent line to . (In case you don’t recognize it, this tangent equation uses the intercept form of a line–a is the x-intercept and b is the y-intercept.) Because the line intersects the parabola in a single point, we can find that point by solving the system of equations defined by the two equations. Substituting for y gives
This is equivalent to , a quadratic. We could determine the value of x using the quadratic formula. Because there is only one solution to this equation (there is only one point of intersection, the point of tangency), the discriminant must be zero. That means
which can be rearranged to give ( is extraneous). Therefore, the area of the triangle formed by the tangent line to and the coordinate axes is no matter what the point of tangency.