How can we minimize this distance?

Question

Given $A = (p, q)$ and $C = (−q, p)$ a pair of points in $\mathbb{R}^2$. Assume that $q > p > 0$.

Find $x, y ∈ R$ such that for $ B = (x, 0), D = (0, y)$, $S = AB + BC − |CD − DA|$ is the smallest. Also, what would be the value of $S$?

Answer 1

$AB+BC$ is independent of |$CD-DA$|, so we can process it separately.

$AB+BC$ need min and it is a typical geo prblem, we mirror A to A'$(p,-q) $,A'C is the min of $AB+BC$,which is $\sqrt{2}(p+q)$

|$CD-DA$| need to be max,$CD-DA=f(y)=\sqrt{q^2+(y-p)^2}-\sqrt{p^2+(y-q)^2}=\dfrac{2(q-p)y}{\sqrt{q^2+(y-p)^2}+\sqrt{p^2+(y-q)^2}}$

$f'(y)=0 \implies y=p+q$, one max point is $\sqrt{2}(q-p)$

we need to check bound also,

$f(+\infty)=q-p,f(-\infty)=p-q$

$S=\sqrt{2}q$