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=k−x Now, the area of the triangle ABC is given by A=12BC⋅AB=12x√(y2−x2)=12x√[(k−x)2−x2] Let u=A2=14x2(k2−2kx) ⇒dudx=12k(kx−3x2) and d2udx2=12k(k−6x) For maximum or minimum of u,dudx=0⇒x=k3(∵x≠0) when x=k3,d2udx2=−12k2<0 ⇒u i.e., A is maximum when x=k3 and y=k−x=2k3. Now, cosθ=BCAC=xy=12⇒θ=π3. Hence, the required angle is π3.

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