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Question

If the sum of the lengths of the hypotenuse and one side of a right-angled triangle is given, the area of the triangle is maximum when the angle between these sides is

A
60
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B
90
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C
30
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D
120
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Solution

The correct option is A 60
Let ABC be a right-angled triangle in which side BC=x(say) and hypotenuse AC=y(say).
Given x+y=k(const.)
y=kx
Now, the area of the triangle ABC is given by
A=12BCAB=12x(y2x2)=12x[(kx)2x2]
Let u=A2=14x2(k22kx)
dudx=12k(kx3x2) and d2udx2=12k(k6x)
For maximum or minimum of u,dudx=0x=k3(x0)
when x=k3,d2udx2=12k2<0
u i.e., A is maximum when x=k3 and y=kx=2k3.
Now, cosθ=BCAC=xy=12θ=π3.
Hence, the required angle is π3.

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