- Home
- Standard 11
- Mathematics
Trigonometrical Equations
normal
In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $P Q, Q R$ and $R P$ at $N, L$ and $M$ respectively, such that the lengths of $P N, Q L$ and $R M$ are consecutive even integers. Then possible length$(s)$ of the side$(s)$ of the triangle is (are)
$(A)$ $16$ $(B)$ $18$ $(C)$ $24$ $(D)$ $22$
A
$(A,D)$
B
$(B,D)$
C
$(B,C)$
D
$(A,C)$
(IIT-2013)
Solution
$\cos P=\frac{(2 n+2)^2+(2 n+4)^2-(2 n+6)^2}{2(2 n+2)(2 n+4)}=\frac{1}{3} $
$\Rightarrow \frac{4 n^2-16}{8(n+1)(n+2)}=\frac{1}{3} $
$=\frac{n^2-4}{2(n+1)(n+2)}=\frac{1}{3} \quad \Rightarrow \quad \frac{n-2}{2(n+1)}=\frac{1}{3} $
$=3 n-6=2 n+2 $
$\Rightarrow n=8 $
$\Rightarrow 2 n+2=18 $
$\Rightarrow 2 n+4=20 $
$\Rightarrow 2 n+6=22$
Standard 11
Mathematics
