Conjugate Tracker

Before you tackle this problem see Root Tracker.

In this problem you must first observe the path of the roots of the quadratic equations $x^2 + px + q = 0$ as you change $p$ and keep $q$ fixed.

You can change the equation $x^2 + px +q = 0$ by moving the point $(p, q)$ in the red frame. You can see how the graph of $y=x^2 + px + q$ changes in the blue frame. The Argand Diagram in the green frame shows the roots of the quadratic equation. Look for two roots in the Argand diagram and watch them move as you change the driving point $(p,q)$ in the red frame, and in doing so change the quadratic equation and its roots.

What do you notice about the paths that these roots follow when you change $p$ and keep $q$ fixed? Make a conjecture about the curves on which the complex roots lie.

Now prove your conjecture.