If in A B C, a=6, b=3 and (A-B) =4 5 , then its area will be - Vidyadip

MCQ

If in $\triangle A B C, a=6, b=3$ and $\cos (A-B)$ $=\frac{4}{5}$, then its area will be

A
7 square units
B
8 square units
✓
9 square units
D
10 square units

✓

Answer

Correct option: C.

9 square units

(C) Let $t =\tan \left(\frac{ A - B }{2}\right)$
$\cos (A-B)=\frac{1-t^2}{1+t^2} \Rightarrow \frac{4}{5}=\frac{1-t^2}{1+t^2} \Rightarrow t=\frac{1}{3}$
So, $\tan \left(\frac{ A - B }{2}\right)=\frac{1}{3}$
Then, $\tan \left(\frac{ A - B }{2}\right)=\frac{ a - b }{ a + b } \cot \frac{ C }{2}$
$\Rightarrow \frac{1}{3}=\frac{6-3}{6+3} \cot \frac{C}{2} \Rightarrow C=90^{\circ}$
$\therefore \quad \Delta=\frac{1}{2}(6)(3) \sin 90^{\circ}=9$ square units.

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