Two straight roadsOA and OBintersect at O -A tower are situated within the angles formed by them and subtend angles of45- and 30- at pointsA andB where the roads are nearest to it- If OA - - and OB - b- then the height of the tower is

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Angle of Elevation

Two straight ...

Question

Two straight roads $O A$ and $O B$ intersect at $O$ .A tower are situated within the angles formed by them and subtend angles of $45^{\circ}$ and $30 °$ at points $A$ and $B$ where the roads are nearest to it. If $O A = α$ and $O B = b,$ then the height of the tower is

$\sqrt{\frac{(a^{2} + b^{2})}{2}}$

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$\sqrt{(a^{2} + b^{2})}$

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$\sqrt{\frac{(a^{2} - b^{2})}{2}}$

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$\sqrt{(a^{2} - b^{2})}$

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Solution

The correct option is C
$\sqrt{\frac{(a^{2} - b^{2})}{2}}$

The explanation for the correct answer :
Solve for the value of height,
Let $P Q$ be the tower of height $h .$
Let $P A,$ and $P B$ be perpendicular from $P$ upon $O A$ & $O B$ respectively.
Then $∠ P B Q = 30^{\circ}$
$O A = a, O B = b \frac{P A}{h} = c o t 45^{\circ} = 1 P A = h \frac{P B}{h} = c o t 30^{\circ} = \sqrt{3} P B = \sqrt{3} h$
Now $O P^{2} = P A^{2} + O A^{2} = P B^{2} + O B^{2}$
$h^{2} + a^{2} = 3 h^{2} + b^{2} \Rightarrow 2 h^{2} = a^{2} - b^{2} \Rightarrow h = \sqrt{\frac{(a^{2} - b^{2})}{2}}$
Hence, option(C) is the correct answer.

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Two straight roadsOA and OBintersect at O .A tower are situated within the angles formed by them and subtend angles of45∘ and 30° at pointsA andB where the roads are nearest to it. If OA=α and OB=b, then the height of the tower is

Two straight roads $O A$ and $O B$ intersect at $O$ .A tower are situated within the angles formed by them and subtend angles of $45^{\circ}$ and $30 °$ at points $A$ and $B$ where the roads are nearest to it. If $O A = α$ and $O B = b,$ then the height of the tower is