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)

JEE Advanced · 2013 · 02 Jun 2013 · Paper 2 · Properties of Triangle

In a triangle

P QR, P

is the largest angle and

cos P = \frac{1}{3}

. Further the incircle of the triangle touches the sides

P Q, QR

and

R P

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)

Answer: (B), (D)

Let the tangent lengths be

P N = 2 x

Q L = 2 x + 2

, and

R M = 2 x + 4

(consecutive even integers).

Since tangents from an external point to a circle are equal:

The sides are:

Using the cosine rule with

cos P = 1/3

cos P = \frac{( 4 x + 2 ) ^{2} + ( 4 x + 4 ) ^{2} - ( 4 x + 6 ) ^{2}}{2 ( 4 x + 2 ) ( 4 x + 4 )} = \frac{1}{3}

Simplifying:

\frac{16 x ^{2} - 16}{2 ( 16 x ^{2} + 24 x + 8 )} = \frac{1}{3}

\frac{4 x ^{2} - 4}{8 x ^{2} + 12 x + 4} = \frac{1}{3}

12 x^{2} - 12 = 8 x^{2} + 12 x + 4

4 x^{2} - 12 x - 16 = 0

x^{2} - 3 x - 4 = 0

(x - 4) (x + 1) = 0

x = 4

(taking positive value).

The sides are:

4 (4) + 2 = 18

4 (4) + 4 = 20

4 (4) + 6 = 22

.

Therefore, possible lengths are 18 and 22.

JEE Advanced · 2013 · 02 Jun 2013 · Paper 2 · One or more correct options