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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

If A, B are the roots of the quadratic equation a b x^2-c^2 x+a b=0...

If tan⁡A,tan⁡B\tan A, \tan B are the roots of the quadratic equation abx2−c2x+ab=0a b x^2-c^2 x+a b=0 where a,b,ca, b, c are sides of the △ABC\triangle A B C then sin⁡2A+sin⁡2B+sin⁡2C\sin ^2 A+\sin ^2 B+\sin ^2 C is

Options

  1. A11
  2. B22
  3. C33
  4. D32\frac{3}{2}

Correct answer

B. 22

Step-by-step solution

Here, tan⁡A+tan⁡B=c2ab\tan \mathrm{A}+\tan \mathrm{B}=\frac{\mathrm{c}^2}{\mathrm{ab}} and tan⁡A⋅tan⁡B=1tan⁡A⋅tan⁡B=1⇒tan⁡A=cot⁡B⇒A=90∘−B⇒A+B=90∘⇒C=90∘⇒sin⁡A=ac,sin⁡B=bc∴sin⁡2A+sin⁡2B+sin⁡2C=a2c2+b2c2+1=2\tan \mathrm{A} \cdot \tan \mathrm{B}=1\tan A \cdot \tan B=1 \Rightarrow \tan A=\cot B \Rightarrow A=90^{{}^{{}^{\circ}}}-B \Rightarrow A+B=90^{{}^{{}^{\circ}}} \Rightarrow C=90^{{}^{{}^{\circ}}}\Rightarrow \sin A=\frac{a}{c}, \sin B=\frac{b}{c}\therefore \sin ^2 A+\sin ^2 B+\sin ^2 C=\frac{a^2}{c^2}+\frac{b^2}{c^2}+1=2
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