Question

In a $\Delta ABC$, if $A=\frac{\pi }{4}$ and $\tan B\tan C=k$
then $k$ must satisfy

• A. $k^2-6k+1\ge 0$
• B. $k^2-6k+1=0$
• C. $k^2-6k+1\le 0$
• D. $3-2\sqrt{2}<k$

Answer 1

The correct option is A $k^2-6k+1\ge 0$

In $\Delta ABC$
$\tan A+\tan B+\tan C=\tan A\tan B\tan C\Rightarrow \tan B+\tan C=\tan A(\tan B\tan C-1)\Rightarrow \tan B+\tan C=\tan \frac{\pi }{4}\times (k-1)=(k-1)\Rightarrow \tan B+\frac{k}{\tan B}=(k-1)\Rightarrow {\tan }^{2}B-(k-1)\tan B+k=0$
For real value of $\tan B$,
$D\ge 0(k-1{)}^2-4k\ge 0k^2-6k+1\ge 0$