Question

Two points at distance x and y from the base point are on the same side of the line passing through the base pf a tower. The angle of elevation from these two points to the top of the tower are complementary. Then, the height of the tower is :

• A. $\sqrt{x}$
• B. $\sqrt{y}$
• C. $\sqrt{xy}$
• D. $\sqrt{\frac{x}{y}}$

Answer 1

The correct option is C $\sqrt{xy}$

Let the tower be AB and C and D be points on the same side

of the tower such that BD = x and BC = y
$\therefore CD=(y-x)$

Also, let $\angle ADB=\theta$ then $\angle ACB=90-\theta$

From $△$ADB, we have tan $\theta$ = $\frac{AB}{DB}=\frac{AB}{x}$ .....1

From $△ABC$, we have $\tan (90-\theta$) = $\frac{AB}{BC}=\frac{AB}{y}$

$\therefore \cot \theta =\frac{AB}{y}$ ..... 2($\because$ tan(90 - $\theta$) = cot $\theta$)

From 1 and 2
$\tan \theta \times cot\theta =1$

$\therefore$$\frac{AB}{x}\times \frac{AB}{y}=1\Rightarrow A{B}^2=xy\Rightarrow AB=\sqrt{xy}$

$\therefore$ The height of the tower is $\sqrt{xy}$