An object is obseved from the points A, B and C lying in a...

An object is obseved from the points

A, B

and C lying in a horizontal straight line which passes directly underneath the object. The angular elevation at

\mathrm{B}

is twice that at

\mathrm{A}

and at

\mathrm{C}

three times that at

\mathrm{A}.

\mathrm{AB}=a, \mathrm{BC}=b

, then the height of the object is An object is obseved from the points A, B and C lying in a horizontal straight line which passes directly underneath

An object is obseved from the points A, B and C lying in a horizontal straight line which passes directly underneath

Options

A $\frac{b}{2 a} \sqrt{(a+b)(3 b-a)}$
B $\frac{a}{b} \sqrt{(a+b)(3 b-a)}$
C $\frac{\mathrm{a}}{2 \mathrm{ b}} \sqrt{(\mathrm{a}+\mathrm{b})(3 \mathrm{ b}-\mathrm{a})}$
D $\frac{2 a}{b} \sqrt{(a+b)(3 b-a)}$

Correct answer

C. $\frac{\mathrm{a}}{2 \mathrm{ b}} \sqrt{(\mathrm{a}+\mathrm{b})(3 \mathrm{ b}-\mathrm{a})}$

Step-by-step solution

Let

\mathrm{ED}=h, \angle \mathrm{EAB}=\alpha\therefore \quad \angle \mathrm{EBD}=2 \alpha, \angle \mathrm{ECD}=3 \alpha

Now,

\angle \mathrm{DBE}=\angle \mathrm{EAB}+\angle \mathrm{BEA}\Rightarrow \quad 2 \alpha=\alpha+\angle \mathrm{BEA}\Rightarrow \quad \angle \mathrm{BEA}=\alpha=\angle \mathrm{EAB}\Rightarrow

\mathrm{AB}=\mathrm{EB}

=a

Similarly,

\angle \mathrm{EBC}=\alpha\operatorname{In} \Delta \mathrm{EBC}, \frac{\mathrm{BC}}{\sin \alpha}=\frac{\mathrm{EB}}{\sin \left(180^{{}^{\circ}}-3 \alpha\right)}\Rightarrow \frac{b}{\sin \alpha}=\frac{a}{\sin 3 \alpha} \Rightarrow \frac{a}{b}=\frac{\sin 3 \alpha}{\sin 3 \alpha}\Rightarrow \quad \frac{a}{b}=\frac{3 \sin \alpha-4 \sin ^{3} \alpha}{\sin \alpha}=3-4 \sin ^{2} \alpha\Rightarrow 4 \sin ^{2} \alpha=3-\frac{a}{b}=\frac{3 b-a}{b}\Rightarrow \sin \alpha=\sqrt{\frac{3 b-a}{4 b}}

\Delta \mathrm{EBD}, \sin 2 \alpha=\frac{E D}{E B}\Rightarrow \quad

\mathrm{ED}=\mathrm{a}

\cdot 2 \sin \alpha \cdot \cos \alpha\Rightarrow h=2 a \sqrt{\frac{3 b-a}{4 b}}, \sqrt{1-\frac{3 b-a}{4 b}}=2 a \sqrt{\frac{3 b-a}{4 b}} \sqrt{\frac{b-a}{4 b}}=\frac{a}{2 b} \sqrt{(a+b)(3 b-a)}

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