A point on the hypotenuse of a right angled triangle is at distance a and b from the sides making right angle, (a,b constant). then hypotenuse has minimum length (a23+b23)23.
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The correct option is A True Let 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.
We have AC=√x2+y2 Ref. image
∴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
Therefore by second derirative test, the length of the hypotenuse is the maximum when
Now when tanθ=(ba)1/3 we have :
Hence the maximum length of the hypotenuses is (a2/3+b2/3)3/2.