Finding the coordinates of a point

Question

I have the $x$ and $y$ values to $2$ points $A(x_a,y_a)$ and $B(x_b,y_b)$. I need to determine the coordinates of a third point $S(x_s,y_s)$ that is at $r$ distance away from both A and B points and lies on a line that is perpendicular to the line $AB$ in the midpoint of $AB$. Also picture because it explains it better than words:

Thank you for the help.

Answer 1

Note that you already know the co-ordinates of the mid-point of the line $AB$ joining $(x_a,y_a)$ and $(x_b,y_b)$. Call this point $D$. So, the co-ordinates of $D$ are $\left(\dfrac{x_a+x_b}{2},\dfrac{y_a+y_b}{2}\right)$.

Also note that $SD \perp AB$. Therefore, the product of the slope of $SD$ and AB is $-1$.

Slope of $AB$=$\dfrac{y_b-y_a}{x_b-x_a}$

So, this gives us the slope of $SD$ and we know a point that lies on it. So, we get the equation of the line $SD$.

The equation of the line $SD$ is $y-\dfrac{y_a+y_b}{2}=\dfrac{x_a-x_b}{y_b-y_a}\cdot\left(x-\dfrac{x_a+x_b}{2}\right)$

Now write the parametric equation of the line $SD$ with respect to the parameter $d$, distance of the point from $D$ and get hold of the co-ordinate $(x_s,y_s)$, by setting $d=\dfrac{\sqrt{r^2-l^2}}{2}$.

Alternatively, note that $SA=SB$ as $S$ is on the perpendicular bisector of $AB$. This coupled with the equation of the line will give you the co-ordinates of $S$.

I'll work the explicit details if you still have problem.