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Question
If the angles of elevation of a tower from two points distant a and b (a>b) from its foot and in the same straight line from it are 30° and 60°, then the height of the tower is
Options
\[\sqrt{a + b}\]
\[\sqrt{ab}\]
\[\sqrt{a - b}\]
\[\sqrt{\frac{a}{b}}\]
Solution
Let h be the height of tower AB
Given that: angle of elevation are`∠C=60°` and`∠D=60°` .`∠D=30°`
Distance `BC=b` and`BD=a`
Here, we have to find the height of tower.
So we use trigonometric ratios.
In a triangle `ABC`,
`⇒ tan C=( AB)/(BC)`
`⇒ tan 60°=(AB)/(BC)`
`⇒ tan 60°= h/b`
Again in a triangle ABD,
`⇒ tan D=(AB)/(BD)`
`⇒ tan 30°=h/a`
Again in a triangle ABD,
`⇒ tan D=(AB)/(BD)`
`⇒ tan 30°=h/a`
`⇒ tan (90°-60°)=h/a`
`⇒ cot 60°=h/a`
`⇒ 1/tan 60°=h/a`
`⇒ b/h=h/a` ` "put tan "60°=h/b`
`⇒ h^2=ab`
`⇒h=ab`
`h=sqrt(ab)`
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