Let $y^2 = x^3 + Ax + B$ be a curve and $y = m(x - x_1) + y_1$ tangent at $x_1$. Why is $x_1$ then a double root?

Question

Suppose we have a function $y^2 = x^3 + Ax + B$ which we differentiate implicit to find $$\frac {dy} {dx} = \frac {3x^2 + A} {2y}$$

Now suppose we know a point $(x_1,y_1)$ on the curve. Define $$y = m(x - x_1) + y_1$$

Write $$(m(x - x_1) + y_1)^2 = x^3 + Ax + B$$ Why is $x_1$ then a double root of this equation ?

The question arises in Eliptic Curve theory. I see that our line is tangent to the curve, but no more.

Is it always true that if a line $L$ defined by $y = m(x-x_1) + y_1$ is tangent at $x_1$ to the curve $y = f(x)$ then for $ m(x-x_1) + y_1 = f(x)$ we have $x_1$ a double root ? Should $y$ be squared and does it also hold for $y^k$ where $k > 2$ ?

Answer 1

You are asking why $f=x^2 - 2x_1x+x_1^2$ divides the difference of the right and left of your equation. Equivalently, why $\,f$ divides $2my_1(x-x_1) +y_1^2-x^3-Ax-B$. But $2my_1=3x_1^2+A$, and $y_1^2 = x_1^3+Ax_1+B$. I think you can do the rest.