Question 4
Positive real numbers $a$, $b$, $c$, $d$ satisfy:
$$
a(c^2 - 1) = b(b^2 + c^2) \text{ and } d \le 1.
$$
Prove that
$$
d(a\sqrt{1-d^2} + b^2\sqrt{1+d^2}) \le \frac{(a+b)c}{2}.
$$
$$
a(c^2 - 1) = b(b^2 + c^2) \text{ and } d \le 1.
$$
Prove that
$$
d(a\sqrt{1-d^2} + b^2\sqrt{1+d^2}) \le \frac{(a+b)c}{2}.
$$