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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2017 · Mathematics · Application of Derivatives

What is the maximum area of a triangle that can be inscribed in a...

What is the maximum area of a triangle that can be inscribed in a circle of radius a?

Options

  1. A3a24\frac{3 \mathrm{a}^2}{4}
  2. Ba22\frac{\mathrm{a}^2}{2}
  3. C33a24\frac{3 \sqrt{3} \mathrm{a}^2}{4}
  4. D3a24\frac{\sqrt{3} a^2}{4}

Correct answer

C. 33a24\frac{3 \sqrt{3} \mathrm{a}^2}{4}

Step-by-step solution

For area of triangle to be maximum, it should be equilateral triangle. Area of △OAB=12absin⁡θ\triangle \mathrm{OAB}=\frac{1}{2} \mathrm{ab} \sin \theta
=12⋅a⋅a⋅sin⁡θ=12a2⋅sin⁡120∘=34a2=\frac{1}{2} \cdot a \cdot a \cdot \sin \theta=\frac{1}{2} \mathrm{a}^2 \cdot \sin 120^{{}^{\circ}}=\frac{\sqrt{3}}{4} \mathrm{a}^2 Area of triangle OAB,OBC,OAC=\mathrm{OAB}, \mathrm{OBC}, \mathrm{OAC}= Area of triangle ABC =3×34a2=334a2\begin{aligned} & =3 \times \frac{\sqrt{3}}{4} a^2 \\ & =\frac{3 \sqrt{3}}{4} a^2 \end{aligned}
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