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

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

A

`lamdalt4/3`

B

`lamdagt5/3`

C

`lamdain((4)/(3),(5)/(3))`

D

`lamdain((1)/(3),(5)/(3))`

Text Solution

Verified by Experts

The correct Answer is:
A
Since, roots are real therefore `Dge0`
`implies4 (a+b+c)^(2)-12lamda(ab+bc+ca)ge0`
`implies(a+b+c)^(2)ge3 lamda(ab+bc+ca)`
`impliesa^(2)+b^(2)+c^(2)ge(ab+bc+ca)(3lamda-2)`
`implies3lamda-2le(a^(2)+b^(2)+c^(2))/(ab+bc+ca)" "...(i)`
Also, `cosA=(b^(2)+c^(2)-a^(2))/(2bc)lt1`
`impliesb^(2)+c^(2)-a^(2)lt2bc`
Similarly, `c^(2)+a^(2)-b^(2)lt2ca`
`and a^(2)+b^(2)-c^(2)lt2ab`
`impliesa^(2)+b^(2)+c^(2)lt2(ab+bc+ca)`
`implies(c^(2)+b^(2)+c^(2))/(ab+bc+ca)=" "...(ii)`
From Eqs. (1) and (ii). we get
`3 lamda-2lt2implieslamdalt(4)/(3)`
Promotional Banner

Similar Questions

Explore conceptually related problems

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

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

Let a ,b , c be the sides of a triangle, where a!=b!=c and lambda in R . If the roots of the equation x^2+2(a+b+c)x+3lambda(a b+b c+c a)=0 are real. Then a. lambda<4/3 b. lambda>5/3 c. lambda in (1/3,5/3) d. lambda in (4/3,5/3)

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

Let a, b, c in R with a gt 0 such that the equation ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0 has non-real roots. If P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b , then

If sinthetaa n d-costheta are the roots of the equation a x^2-b x-c=0 , where a , ba n dc are the sides of a triangle ABC, then cosB is equal to 1-c/(2a) (b) 1-c/a 1+c/(c a) (d) 1+c/(3a)

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 , c are positive numbers such that a gt b gt c and the equation (a+b-2c)x^(2)+(b+c-2a)x+(c+a-2b)=0 has a root in the interval (-1,0) , then

Let a , ba n dc be the three sides of a triangle, then prove that the equation b^2x^2+(b^2=c^2-alpha^2)x+c^2=0 has imaginary roots.

If the sines of the angles A and B of a triangle ABC satisfy the equation c^2x^2-c(a+b)x+a b=0 , then the triangle a)acute angled b)right angled c)obtuse angled d) sinA+cosA = ((a+b))/c