Parametric Equations and the S-Notation

Every point on $y^2=4ax$ can be written in a single-parameter form that is often far more convenient than working with $x$ and $y$ separately. Check that the point $(at^2, 2at)$ satisfies the equation for every real $t$: substituting gives $(2at)^2 = 4a(at^2)$, i.e. $4a^2t^2=4a^2t^2$, which is an identity. Conversely, given any point $(x,y)$ on the curve with $x\ge 0$, setting $t=y/(2a)$ recovers $x=at^2$ automatically. So

are the parametric equations of the parabola, and the point $(at^2,2at)$ is often just called "the point $t$", written $P(t)$. This single real number $t$ is a compact stand-in for an entire point on the curve, and tangent/normal formulas both have unusually clean forms when written in terms of $t$ — reason enough to get comfortable with it early.

It is also useful to fix some shorthand notation before writing down tangents, normals, and chords, since the same three expressions appear over and over. For the parabola $S \equiv y^2-4ax=0$ and points $(x_1,y_1)$, $(x_2,y_2)$ in the plane, define

$S_1$ is obtained from $S$ by the substitution $y^2 \to yy_1$ and $2x \to x+x_1$ (a pattern worth memorising, since it recurs for every conic); $S_{11}$ is just $S$ evaluated at $(x_1,y_1)$ itself, and it tells you exactly where a point sits relative to the curve. Writing $M=(x_1,0)$ and letting $Q=(x_1, 2\sqrt{ax_1})$ be the point directly above/below $M$ that is on the parabola, a short comparison of $(MP)^2=y_1^2$ against $(MQ)^2=4ax_1$ shows:

• $P(x_1,y_1)$ lies outside the parabola $\iff y_1^2 > 4ax_1 \iff S_{11}>0$.
• $P(x_1,y_1)$ lies on the parabola $\iff y_1^2=4ax_1 \iff S_{11}=0$.
• $P(x_1,y_1)$ lies inside the parabola (on the focus's side) $\iff y_1^2<4ax_1 \iff S_{11}<0$.

This $S_{11}$ test is the standard way to check, without drawing anything, whether a given point is inside, outside, or exactly on a parabola — and it is the first ingredient in deciding, later, how many tangents can be drawn to the curve from that point. …