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On one side of a road of width $d$ meters there is a point of observation $P$ at a height $h$ meters from the ground. If a tree on the other side of the road, makes a right angle at $P$ , height of the tree in meters is

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\frac{h^{2} - d^{2}}{h}

\frac{h^{2} + d^{2}}{h}

\frac{d^{2} - h^{2}}{h}

\frac{2 d^{2} + h^{2}}{h}

answer is B.

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Detailed Solution

It is given that on one side of a road of width

d

meters there is a point of observation P and a tree on the other side of the road, makes a right angle at

P

.
According to the given data, the figure has been drawn below.
Question Image

△ PBA,

{AP}^{2} = h^{2} + d^{2}

(where h is the height of PB and d is the length of BA.)
In

△ PCD,

{PC}^{2} = {(H - h)}^{2} + d^{2}

(where H is the height of CA and DA = PB = h meter,
therefore, CD =

H - h

)
In

△ ADC,

{AC}^{2} = {AP}^{2} + {PC}^{2}

{\Rightarrow H}^{2} = h^{2} + d^{2} + {(H - h)}^{2} + d^{2}

{\Rightarrow H}^{2} = h^{2} + 2 d^{2} + H^{2} + h^{2} - 2 Hh

\Rightarrow 2 Hh = 2 h^{2} + 2 d^{2}

\Rightarrow Hh = h^{2} + d^{2}

\Rightarrow H = \frac{h^{2} + d^{2}}{h}

So, height of the tree in meters is

\frac{h^{2} + d^{2}}{h}

.
So the correct option is 2.

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