Question

A vertical pole $PS$ has two marks at $Q$ and $R$ such that the portions $PQ,PR$ and $PS$ subtend angles $\alpha ,\beta$ and $\gamma$ at a point on the ground distance $x$ from the bottom of pole. If $PQ=a,PR=b,PS=c$ and $\alpha +\beta +\gamma ={180}^{\circ }$ then $x^2$ is equal to

• A. $\frac{a^3}{a+b+c}$
• B. $\frac{b^3}{a+b+c}$
• C. $\frac{c^3}{a+b+c}$
• D. $\frac{abc}{a+b+c}$

Answer 1

Answer: (D)

$\frac{abc}{a+b+c}$

We have $\tan \alpha =\frac{a}{x},\tan \beta =\frac{b}{x}$ and $\tan \gamma +\frac{c}{x}$
$\therefore \alpha +\beta +\gamma ={180}^{\circ }$
So, $\tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \cdot \tan \beta \cdot \tan \gamma$
or $\frac{a}{x}+\frac{b}{x}+\frac{c}{x}=\frac{a}{x}\cdot \frac{b}{x}\cdot \frac{c}{x}$
or $x^2=\frac{abc}{a+b+c}$.