Q.Find the equation of tangent to the parabola $y^2 = 36x$ from the point $(2,9)$.

A tangent to a parabola is the limiting position of a secant line as its two points of intersection with the curve merge into one — equivalently, a line touching the curve at exactly one point without crossing it there. Differentiating the standard equation $y^2=4ax$ gives the slope $dy/dx=2a/y$ at any point, which leads to the point-form tangent at $(x_1,y_1)$: $yy_1=2a(x+x_1)$ — obtained from the original equation by the simple substitution …