Find the perpendicular distance between 2 lines

Question

I'm trying to find an equation $d(x,y)$ that defines the distance between a line with slope $1/0.4$ (let's call it $L_1$) and a line with slope $1/2.25$ (let's call it $L_2$). The distance equation must be perpendicular to line $L_1$ but that's the only restriction. The two lines intersect at the point $(x,y) = (762,0)$. Any help would be appreciated, I'm not too sure how to approach this.

Thanks,

Felix

Answer 1

Given a point $(x_0, y_0)$ on $L_1$. The line perpendicular to $L_1$ has slope $=-0.4$. Therefore the equation of the perpendicular line passing through $(x_0,y_0)$ is given by:

$$\frac{y-y_0}{x-x_0} = -0.4$$

Let's call the line described above as $L_3$.

Now if we calculate the intersection of $L_3$ and $L_2$, say it comes out to be $(x_1, y_1)$. Then we can compute the distance between these two points by applying the formula:

$$d = \sqrt{(x_0-x_1)^2 + (y_0-y_1)^2}$$