Point of Perpendicular Intersection on Real Line from Points in Complex Plane/Examples/a = 1-3i, b = -3+4i

Examples of Point of Perpendicular Intersection on Real Line from Points in Complex Plane
Let $a, b \in \C$ be complex numbers represented by the points $A$ and $B$ respectively in the complex plane.

Let $x \in \R$ be a real number represented by the point $X$ on the real axis such that $AXB$ is a right triangle with $X$ as the right angle. Let $a = 1 - 3 i, b = -3 + 4 i$.

The point $X$ on the positive half of the real axis is at:
 * $x = 3$

Proof
From Point of Perpendicular Intersection on Real Line from Points in Complex Plane:


 * $x = \dfrac {a_x + b_x \pm \sqrt {a_x^2 + b_x^2 - 2 a_x b_x - 4 a_y b_y} } 2$

Setting $a = 1 - 3 i, b = -3 + 4 i$:

It is the positive solution that is needed, so:


 * $x = 3$