Parametric Form

Point at signed distance r from (x₁, y₁) along the line of inclination θ. r > 0 in the direction of increasing θ, r < 0 in the opposite direction.

Let a line pass through $A(x_1,y_1)$ at inclination $\theta$. A point $P(x,y)$ on this line at signed distance $r$ from $A$ satisfies:

$$x - x_1 = r\cos\theta, \qquad y - y_1 = r\sin\theta$$

since the displacement from $A$ to $P$ has magnitude $|r|$ in direction $(\cos\theta, \sin\theta)$.

Dividing:

$$\boxed{\frac{x-x_1}{\cos\theta} = \frac{y-y_1}{\sin\theta} = r}$$

$r > 0$: point is in the forward direction ($\theta$ increasing). $r < 0$: backward direction.

Why this matters

The parametric form converts the line into a number line. Every point corresponds to one value of $r$, and $|r|$ is its distance from $A$.

Point at given distance: set $|r|$ to the required value. Two solutions ($\pm r$) give two points equidistant from $A$ on opposite sides.

Chord cut by a curve: substitute $(x_1 + r\cos\theta,\; y_1 + r\sin\theta)$ into the curve equation. The result is a quadratic in $r$. Its roots $r_1, r_2$ give the signed distances to the two intersection points. Length of chord $= |r_1 - r_2|$; midpoint corresponds to $r = \frac{r_1+r_2}{2}$ — obtainable from Vieta's without solving the quadratic.

> > $$ > \vec{r} = \vec{a} + \lambda\,\hat{d} > $$ > > where $\lambda = r$ is the signed distance parameter. In 3D this generalises immediately to $\vec{r} = \vec{a} + \lambda\vec{b}$ where $\vec{b}$ need not be a unit vector — $\lambda$ then loses its distance interpretation but the line is the same. The 2D parametric form is the bridge between coordinate geometry and vector geometry of lines.