How To: Find a Point on a Line Given a Point and a Distance

Posted On 11/12/2016

I faced a little mathematical challenge recently. It is as follow:

Given a line with a known equation, a point $(x_a, y_a)$ and a distance $d$ , find the coordinates of the point $(x_b, y_b)$ that is at a distance $d$ from $(x_a, y_a)$ .

The equation of the line is:

$y=mx+c$

Which is equivalent to:

$y=m(x-x_0)+y_0$

The trick here is to use the circle formula:

$x^2+y^2=r^2$

In our case, the center of the circle is $(x_a, y_a)$ and the radius is the distance $d$ . Using this formula, we will be able to find $x_b$ . After that, it is as simple as using the line formula to find $y_b$ .

Let’s start by finding $x_b$ .

$x_b^2+y_b^2=r^2$
$(x_b-x_a)^2+(y_b-y_a)^2=d^2$
$(x_b-x_a)^2+m^2(x_b-x_a)^2=d^2 \text{ (Replace with the line formula)}$
$(x_b-x_a)^2(1+m^2)=d^2$
$(x_b-x_a)^2=d^2/(1+m^2)$
$x_b = x_a + d/\sqrt(1+m^2)$