Question

Assertion (A)
If two towers of height h₁ and h₂ subtend angles of 60° and 30° respectively at the midpoint of the line joining their feet, then h₁ : h₂ = 3 : 1.

Reason (R)
Figure
In the given figure, we have: $\frac{h_1}{x}=\tan 60°\mathrm{and}\frac{h_2}{x}=\tan 30°$

(a) Both Assertion (A) and Reason (R) are true and (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Answer 1

There are two towers $AB$ and $CD$ of heights $h_1$ and $h_2$, respectively. Let $AC$ be the distance between the two towers and O be the midpoint of $AC$.
Thus, we have:
$AO=CO=x$ m
Now, in ∆$AOB$, we have:
$\frac{AB}{AO}=\tan {60}^0=\sqrt{3}$
⇒ $\frac{h_1}{x}=\sqrt{3}$ ⇒ $h_1=x\sqrt{3}$
Again, in ∆$DOC$, we have:
$\frac{DC}{OC}=\tan {30}^o=\frac{1}{\sqrt{3}}$
⇒ $\frac{h_2}{x}=\frac{1}{\sqrt{3}}$ ⇒ $h_2=\frac{x}{\sqrt{3}}$
Now,
$\frac{h_1}{h_2}=\frac{x\sqrt{3}}{{\displaystyle \frac{x}{\sqrt{3}}}}=\frac{3}{1}$
∴ $h_1:h_2=3:1$

Hence, both assertion (A) and reason (R) are true and (R) is the correct explanation of (A).