Question

Assertion :If in a triangle,
$\tan A:\tan B:\tan C=1:2:3$ then $A={45}^0$ Reason: If $\alpha :\beta :\gamma =1:2:3,$ then $\alpha =1$

• A. Both A and R are individually true and R is the correct explanation of A
• B. Both A and R are individually true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true.

Answer 1

The correct option is C A is true but R is false

Let $\tan A=\lambda ,\tan B=2\lambda ,\tan C=3\lambda$
$\because$ In triangle,
$\tan A+\tan B+\tan C=\tan A\tan B\tan C$
$\Rightarrow \lambda +2\lambda +3\lambda =\lambda \times 2\lambda \times 3\lambda$
$\Rightarrow 6\lambda =6{\lambda }^3$
$\Rightarrow \lambda (1-{\lambda }^2)=0$
$\Rightarrow \lambda =-1,0,1$
$\Rightarrow \lambda =1,\lambda \ne -1,0$
$\therefore \tan A=1\Rightarrow A={45}^0$
If $\lambda =-1$ then all angles becomes obtuse
and if $\lambda =0$ then all angles become zero.
and the reason is obviously wrong.