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Q.

A man standing between two vertical posts finds that the angle subtended at his eyes by the tops of the posts is a right angle. If the height of the two posts are two times and four times the height of the man and the distance between the poles is equal to the height of the largest pole, then the ratio of the distance of the man from the shorter and the longer post is

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a

3 : 1

or 1:3

b

2 : 3

or 3:2

c

3 : 2

or 2:3

d

1 : 5

or 5:1

answer is A.

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Detailed Solution

It is given that the man stands between two vertical posts and he finds the angle subtended at his eyes by the tops of the posts is a right angle. The height of the two posts is two times and four times of the height of the man and the distance between the poles is equal to the height of the largest pole.
According to the given data the figure has been drawn below.
Question Image

Question Image

Let

AB

be the men of height

h m

,

CD

and

EF

be the two towers of height

2 h m

and

4 h m

respectively.

∴ ∠ CAE = 90 °

Draw

PQ ∥ DF

passing through

A

.

\Rightarrow CP = PD = h m, QF = h m

∴ EQ = 3 h m

Let

DB

be

d m

\Rightarrow DB = PA = d m

and

BF = AQ = (4 h - d) m

PAQ is a straight line.

\Rightarrow ∠ PAC + ∠ CAE + ∠ EAQ = 180 °

\Rightarrow ∠ PAC + 90 ° + ∠ EAQ = 180 °

(

∴ ∠ CAE = 90 °)

\Rightarrow ∠ PAC + ∠ EAQ = 90 °

Let

∠ PAC

be

θ

.

\Rightarrow ∠ EAQ = 90^{o} - θ

In

△ CPA,

applying tangent rule,

tan θ = p e r p e n d i c u l a r b a s e

tan θ = C P A P tan θ = h d ...... (1)

In

△ AQE

applying tangent rule,

tan θ = p e r p e n d i c u l a r b a s e

tan 90 ° − θ = E Q A Q tan 90 ° − θ = 3 h 4 h − d cot θ = 3 h 4 h − d ...... (2) ∵ tan 90 ° − θ = cot θ

From eq. (1) and (2),

⇒ cot θ = 3 h 4 h − d ⇒ 1 tan θ = 3 h 4 h − d ∵ 1 tan θ = cot θ ⇒ d h = 3 h 4 h − d from eq . (1)

\Rightarrow 3 h^{2} = d (4 h - d)

\Rightarrow 3 h^{2} = 4 dh - d^{2}

\Rightarrow (d - 3 h) (d - h) = 0

\Rightarrow d - 3 h = 0

or

d = h

if (d- 3h) = 0 then we get

d = 3 h .

Now,

\frac{DB}{BF} = \frac{3 h}{h}

\Rightarrow \frac{DB}{BF} = \frac{3}{1}

\Rightarrow DB : BF = 3 : 1

.
When

d = h

,

\frac{DB}{BF} = \frac{h}{3 h}

\Rightarrow \frac{DB}{BF} = \frac{1}{3}

\Rightarrow DB : BF = 1 : 3

So, the correct option is

1

.

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