How a formula is developed

Question

The rule for converting line equations to parametric equations is:

$$\frac{(x-x_1)}{a} =\frac{(y-y_1)}{b} =t$$

I would like to know how this was developed.

Thank you.

Answer 1

A line is given by a point $p = (x,y)$ and a vector $v = (a,b)$ then a parametrization of this line will be given by $\gamma(\infty, \infty) \to \Bbb{R}^2$ $\gamma(t) = p + t v = (x + ta, y + t b)$.

Note that if $q = (x_1,y_1) = \gamma(t)$ is another point in the line then $$\frac{x_1 - x}{a} = t = \frac{y_1 - y}{b}$$

Answer 2

At first we can parameterize a line $y = mx + c$ by $x$ itself. Then we can think of parameterizing $$y - y_0 = m(x-x_0)$$ by $t = x - x_0$.

For an arbitrary line in the plane--including a vertical line--of the form $a(y-y_0) = b(x-x_0)$, we can parameterize it any number of ways. Setting

$$t = \frac{x - x_0}{a} = \frac{y - y_0}{b}$$ can be convenient. It is naturally equivalent to another representation:

$$\vec r(t) = (x(t), y(t)) = (x_0, y_0) + t(a,b)$$

Answer 3

In 2-D Cartesian system taking a point $P(x_1,y_1)$. now our line make $\theta $ angle with positive x-axis then a point on our locus say $(x,y)$ at a distance t makes component $t\cos{\theta}$ on x -axis hence the x coordinate is $x_1+t\cos\theta$ and similarly $t\sin\theta$ on y axis hence y coordinate is $y_1+t\sin\theta$ and this is what our parametric equation says.so in your case $a=cos\theta $ and $b= sin\theta$