Question

The angle of elevation of a vertical pillar stands on a horizontal plane from any place is ∅. On moving 'a' distance towards the pillar the angle becomes 45° & further moving b distance it becomes (90°-∅). Find the height of pillar.

Answer 1

SOLUTION ⬆️

Refer to the attachment.

Hope it helps ☺️

Answer 2

The height of pillar=$\frac{ab}{a-b}$

Step-by-step explanation:

Let height  of pillar=h

BE=x

DE=a

CD=b

BC=x-(a+b)

BD=x-(a+b)+b=x-a

In triangle ABC

$\theta=\phi$

$tan\theta=\frac{AB}{BC}=\frac{h}{x-(a+b)}$...(1)

By using $\frac{Perpendicular\;side}{Base}=tan\theta$

In triangle ABD

$tan45^{\circ}=\frac{AB}{BD}=\frac{h}{x-a}$

$h=x-a$

Because $tan45^{\circ}=1$

In triangle ABE

$tan(90-\phi)=\frac{h}{x-(a+b)}$

$cot\phi=\frac{h}{x-(a+b)}$..(2)

using$cot\theta=tan(90-\theta)$

Multiply equation (1) by (2)

$tan\theta\times cot\theta=\frac{h^2}{x(x-(a+b))}$

We know that

$cot\theta=\frac{1}{tan\theta}$

Using the formula

$1=\frac{h^2}{x(x-(a+b))}$

$h^2=x(x-(a+b))$

$h^2=(h+a)(h-b)$

$h^2=h^2-bh+ah-ab$

$h^2-h^2+bh-ah=-ab$

$-h(a-b)=-ab$

$h=\frac{-ab}{-(a-b)}=\frac{ab}{a-b}$

Hence, the height of pillar=$\frac{ab}{a-b}$

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