Question

Two trees of heights $a\text{units}$ and $b\text{units}$ which are $c\text{units}$ apart are supported by a rope from some point on the ground between the trees. If the length of rope used is minimum then the distance of that point on the ground from any of the trees can be

• A. $\frac{ab}{a+b}$ units
• B. $\frac{bc}{a+b}$ units
• C. $\frac{ac}{a+b}$ units
• D. $\frac{abc}{a+b}$ units

Answer 1

Answer: (B), (C)

$\frac{ac}{a+b}$ units

From given information :
Let $AP=a,BQ=b;R$ is the grounded point and $PR=x,RQ=c-x$
Now, total rope used $=AR+RB>AB\left(\text{triangle rule}\right)$
So, the sum will be minimum if points $A,R$ and $B$ will be collinear,
$\Rightarrow △APR$ will be similar to $△CQR$
$\Rightarrow \frac{a}{b}=\frac{x}{c-x}$
$\Rightarrow \frac{b}{a}=\frac{c}{x}-1$
$\Rightarrow x=\frac{ac}{a+b}$
and $c-x=\frac{bc}{a+b}$