    Question

# A point on the hypotenuse of a right angledtriangle is at distance a and b from the sides making rightangle, (a,b constant). then hypotenuse hasminimum length (a23+b23)23.

A
True
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
False
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is A TrueLet ΔABC be right angled at & let AB=x and BC=y. Let P be a point in the hypotenuse of the triangle such that P is at a distance of a and b from the sides AB and BC respectively.Let ∠C=θ We have AC=√x2+y2 Ref. imageNowPC=bcosecθAnd, AP=asecθAC=AP+PCAC=bcosecθ+asecθ→ (1)∴d(AC)dθ=−bcosecθ.cotθ+asecθ.cotanθ∴d(AC)dθ=0asecθtanθ=bcosecθ.cotθacosθ.sinθcosθ=bsinθ.cosθsinθ asin2θ=bcos2θ(a)1/3sinθ=(b)1/3cosθtanθ=(ba)13∴sinθ=(b)1/3√a2/3+b2/3 and cosθ=cos1/3√a2/3+b2/3 ___(2)It can be clearly shown that d2(AC)dθ2<0 when tanθ=(ba)13Therefore by second derirative test, the length of the hypotenuse is the maximum whentanθ=(ba)1/3Now when tanθ=(ba)1/3 we have :AC=b√a2/3+b2/3b1/3+a√a2/3+b2/3a1/3=√a2/3+b2/3(b2/3+a2/3)=(a2/3+b2/3)3/2Hence the maximum length of the hypotenuses is (a2/3+b2/3)3/2.   Suggest Corrections  0      Similar questions  Related Videos   Parametric Differentiation
MATHEMATICS
Watch in App  Explore more