Tangent to a Parabola

What is a tangent, geometrically? Picture a secant line cutting the parabola at two points, $P$ and a second point $Q$. Now let $Q$ slide along the curve, getting closer and closer to $P$. As $Q\to P$, the secant line rotates and settles into a single limiting position — that limiting line is called the tangent to the curve at $P$. Equivalently: a tangent is a straight line that meets the parabola in two coincident points (i.e. it "just touches" the curve at one point without crossing through it there).

Finding the tangent at a point, using calculus. We want the equation of the tangent to $y^2=4ax$ at a point $P(x_1,y_1)$ on it.

The general fact from calculus is: the tangent to a curve $y=f(x)$ at $(x_1,y_1)$ has equation $y-y_1=\left[f'(x)\right]_{(x_1,y_1)}(x-x_1)$ — i.e. we need the slope of the tangent, which is $\dfrac{dy}{dx}$ evaluated at $(x_1,y_1)$.

Differentiate the parabola's equation $y^2=4ax$ implicitly with respect to $x$:

At the point $P(x_1,y_1)$, the slope is therefore $\dfrac{2a}{y_1}$. Substituting into the point-slope form:

But $P$ lies on the parabola, so $y_1^2=4ax_1$. Substituting this in:

This is the point-form tangent: $\boxed{yy_1=2a(x+x_1)}$ — note the simple pattern of replacing $y^2\to yy_1$ and $x\to\dfrac{x+x_1}{2}$ in the original equation, a pattern that recurs (with the appropriate substitution) for the ellipse and hyperbola too. …