Consider an ellipse with foci in $(-c, 0)$ and $(c, 0)$, where $a$ is the length of the semimajor axis. Consider a point $(x,y)$ belonging to the ellipse. The point verifies the equation:
$$\sqrt{(x - c)^2 + y^2} = 2a - \sqrt{(x + c)^2 + y^2}$$
The Left Hand Side is also the distance $d_1$ between the point $(x,y)$ and the focus $(c, 0)$. How to obtain the $x$ value which minimizes such a distance?
It should be $x = a$. I tried to use the Right Hand Side as a function and to compute is partial derivative with respect to $x$:
$$\frac{\partial}{\partial x} \left[ 2a - \sqrt{(x + c)^2 + y^2} \right] = - \frac{x + c}{\sqrt{(x + c)^2 + y^2}}$$
If this is algebraically correct, there are two issues:
- it does not depend on $a$;
- the solution $x = -c$ is not even a point of the ellipse.
Is it possible to obtain the abscissa of the ellipse point which minimizes $d_1$ from the above equation? If yes, how? If no, which is the correct procedure?