Home
Class 12
MATHS
If a, b, c are the sides of triangle , t...

If `a`, `b`, `c` are the sides of triangle , then the least value of `(a)/(c+a-b)+(b)/(a+b-c)+(c )/(b+c-a)` is

A

`1//3`

B

`1`

C

`3`

D

`6`

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` `c+a-b`, `b+c-a`,`a+b-c` are all positive
Using `A.M. ge G.M.`
`:.(a)/(c+a-b)+(b)/(a+b-c)+(c )/(b+c-a) ge [(abc)/((c+a-b)(a+b-c)(b+c-a))]^(1//3)`........`(i)`
Also , `a^(2) ge a^(2)-(b-c)^(2)`
`implies a^(2) ge (a+b-c)(a-b+c)`
Similarly `b^(2) ge (b+c-a)(b-c+a)`
`c^(2) ge (c+a-b)(c-a+b)`
`:.a^(2)b^(2)c^(2) ge (a+b-c)^(2)(b+c-a)^(2)(c+a-b)^(2)`
Thus `abc ge (a+b-c)(b+c-a)(c+a-b)`
`implies (abc)/((c+a-b)(a+b-c)(b+c-a)) ge 1`
Hence, from `(i) (abc)/((c+a-b)(a+b-c)(b+c-a)) ge 1`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Comprehension|2 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Examples|37 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE|Exercise Solved Examples And Exercises|222 Videos

Similar Questions

Explore conceptually related problems

If a , b and c are the side of a triangle, then the minimum value of (2a)/(b+c-a)+(2b)/(c+a-b)+(2c)/(a+b-c)i s (a) 3 (b) 9 (c) 6 (d) 1

If a, b and c represent the lengths of sides of a triangle then the possible integeral value of (a)/(b+c) + (b)/(c+a) + (c)/(a +b) is _____

Let a,b,c be the sides of a triangle ABC, a=2c,cos(A-C)+cos B=1. then the value of C is

IF tan A and tan B are the roots of abx^2 - c^2 +ab =0 where a,b,c are the sides of the triangle ABC then the value of sin ^2 + sin ^2 B + sin ^2 C is

Given that a , b , c , are the side of a A B C which is right angled at C , then the minimum value of (c/a+c/b)^2 is

If a ,b ,and c are the sides of a triangles and A , B and C are the angles opposites to a ,b and c , respectively then "Delta"=|a^2b s in A c s in A bs in A1cos A cs in A cos A1| is independent of

If a, b, c are the sides of a scalene triangle such that |[a , a^2 , a^(3)-1 ],[b , b^2 , b^(3)-1],[ c, c^2 , c^(3)-1]|=0 , then the geometric mean of a, b c is

In a triangle ABC , if a,b,c are the sides opposite to angles A , B , C respectively, then the value of |{:(bcosC,a,c cosB),(c cosA,b,acosC),(acosB,c,bcosA):}| is

If a, b, c are the sides of a triangle ABC such that x^2+2(a+b+c)x+3lambda(a b+b c+c a)=0 has real roots.then

In triangle A B C , let /_c=pi/2dot If r is the inradius and R is circumradius of the triangle, then 2(r+R) is equal to (a) a+b (b) b+c (c) c+a (d) a+b+c