A ladder rests against a vertical wall at an angle α to the horizontal. If the foot is pulled away through a distance 2 m , then it slides a distance 5 m down the wall, finally making an angle β with the horizontal. The value of tan ⁡ α + β 2 is equal to

Question

A ladder rests against a vertical wall at an angle α to the horizontal. If the foot is pulled away through a distance 2 m , then it slides a distance 5 m down the wall, finally making an angle β with the horizontal. The value of tan ⁡ α + β 2 is equal to, 2 5, 5 2, 10, None of these

MARKS App · Accepted Answer

Let, the length of the ladder be l and A B is the wall. ∴ A D = E C = l Now, B E - B D = 2 ⇒ l cos ⁡ β - l cos ⁡ α = 2 ⇒ l cos ⁡ β - cos ⁡ α = 2 ⇒ 2 l sin ⁡ α + β 2 sin ⁡ α - β 2 = 2 … … i Also, B A - B C = 5 ⇒ l sin ⁡ α - l sin ⁡ β = l sin ⁡ α - sin ⁡ β = 5 ⇒ 2 l cos ⁡ α + β 2 sin ⁡ α - β 2 = 5 … … ii Dividing equation i & i i , we get, ∴ tan ⁡ α + β 2 = 2 5

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