A) 2 : 1
B) 1 : 3
C) 1 : 5
D) 1 : 2
Correct Answer: D
Solution :
Idea Here, We use Napier's analogy and we know that\[\tan \theta =\frac{1}{\cot \theta }\] Here, given that in \[\Delta ABC\] \[\tan \left( \frac{A-B}{2} \right)=\frac{1}{3}\tan \left( \frac{A+B}{2} \right)\] ?(i) Using Napier's analogy, \[\tan \left( \frac{A-B}{2} \right)=\frac{a-b}{a+b}\cot \left( \frac{C}{2} \right)\] ?(ii) From Eqs. (i) and (ii), we get \[\frac{1}{3}\tan \left( \frac{A-B}{2} \right)=\frac{a-b}{a+b}\cot \left( \frac{C}{2} \right)\] \[\Rightarrow \] \[\frac{1}{3}\tan \left( \frac{C}{2} \right)=\frac{a-b}{a+b}\cot \left( \frac{C}{2} \right)\] \[[\because A+B+C=\pi /2]\] \[\Rightarrow \] \[\frac{a-b}{a+b}=\frac{1}{3}\] \[\Rightarrow \] \[3a-3b=a+b\] \[2a=4b\] \[\Rightarrow \] \[\frac{a}{b}=\frac{2}{1}\Rightarrow \frac{b}{a}=\frac{1}{2}\] Thus, the ratio of the sides opposite to the angle is b : a = 1 : 2 TEST Edge Relation between the sides and angle of triangle, related questions are asked from this concept. Students are advised to understand the proper using of trigonometric identities and stick with concept of properties of triangle.
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