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If A be the area of a right triangle and b one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is 2 Ab /√ b4+4A2.
Ans: Let QR = b
A = Ar(ΔPQR)
A = 1/2 x b x PQ
PQ = 2A/b ...........................(1)
Δ PNQ ∼ ΔPQR (AA)
⇒ PQ/PR = NQ/QR .....................(2)
From Δ PQR
PQ2 + QR2 = PR2
4A2/ b2 + b2 = PR2
Equation (2) becomes
2 A/bxPR = NQ/b
NQ = 2A/PR
NQ = 2 A/√4 A2 + b4
solve: 4ydx+xdy=0
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