Tangent line problem

A secant line intersects a curve at two distinct points and an equation can be found easily using the coordinates of the two points. Letting $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ be two points that lie on a curve then the secant line through these points can be represented algebraically as $y-y_2=\frac{y_2-y_1}{x_2-x_1}x-x_2.$ With a tangent line, we only know one point which is not enough to define a line, which creates difficulties in creating this equation.