Finding the intersection of two "half" hyperbolas with equations of the form $\sqrt{(x-x_1)^2+(y-y_1)^2}-\sqrt{(x-x_2)^2+(y-y_2)^2}=C$

Question

I have two hyperbola equations that need to find an interaction, I tried to use the GeoGebra to draw the graph, but I cannot implement this using C++ as I am not able to convert the two equations like $x = \text{??}$ or $y = \text{??}$.

Is there any software or anyway I can make these two equations to be the format like: $x = \text{??}$ and $y = \text{??}$ ? For your convenience, I will paste the equation here:

$$\begin{align} \sqrt{(x-x_1)^2+(y-y_1)^2}-\sqrt{(x-x_2)^2+(y-y_2)^2}=C_1 \\[6pt] \sqrt{(x-x_1)^2+(y-y_1)^2}-\sqrt{(x-x_3)^2+(y-y_3)^2}=C_2 \end{align}$$

Thank you so much for your help, appreciated

Answer 1

There is an analytical solution to your problem (it is almost half a mile long).

Instead, take the first equation and square it. This gives a quadratic equation in $y(x)$ (two roots). Plug in the second equation, repeat squaring to have a single equation in $x$ and ... have fun !