Question

The sum of the hypotenuse and a side of a right angled triangle is constant. If the area of the triangle is maximum then the angle between the hypotenuse and the given side is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is C $\frac{\pi }{3}$

$h+a=constant=k$
$h^2=a^2+b^2$ (property is right $\Delta$)
area of $\Delta =\frac{1}{2}ab$

$area=\frac{1}{2}a\sqrt{k(h-a)}$
$=\frac{1}{2}a\sqrt{k(k-2a)}$

$\frac{d\left(area\right)}{da}=\frac{1}{2}\sqrt{k(k-2a)}+\frac{1}{2}ak\frac{\left(\frac{-2}{2}\right)}{\sqrt{k(k-2a)}}$
$0=k(k-2a)-ak$
$a=\frac{k}{3}$, $h=\frac{2}{3}k$
$cos\theta =\frac{a}{h}$
$\theta =\frac{\pi }{3}$