Question

If a,b and c are sides of a triangle where a not equal to b not equal to c and

Answer 1

\(x^2+2(a+b+c)x+3 \lambda(ab+bc+ca)=0 \) has real roots then
A) lambda<4/3
B) Lambda>5/3
C)Lambda belongs to (1/3,5/3)
D)Lambda belongs to (4/3,5/3)

Answer 2

Real roots:
\[\large 4(a+b+c)^2-12 \lambda(ab+bc+ca) \geq 0\]
\[\large 3 \lambda \leq \frac{(a+b+c)^2}{(ab+bc+ca)}\]

Answer 3

\[\large 3 \lambda \leq \frac{a^2+b^2+c^2}{ab+bc+ca}+2\]

Answer 4

what now??

Answer 5

@wio @Psymon

Answer 6

a > b > c

Answer 7

wolg we can assume that

Answer 8

how can we assume :/

Answer 9

is there anything like
a-b<c ?

Answer 10

thats true for any triangle difference between two sides must be less than third side,
but hows it useful here ?

Answer 11

a > b > c, comes from scalene thing btw

Answer 12

thats true for any triangle difference between two sides must be less than third side,

is another way of saying,

sum of any two sides must be greater than third side
b+c > a
or c > a-b

Answer 13

well..
\[(a-b)<c\]
\[(a-b)^2<c^2\]
\[a^2+b^2-2ab<c^2\]
\[b^2+c^2-2bc<a^2\]
\[a^2+c^2-2ab<b^2\]
Add them all..
\[2(a^2+b+^2+c^2)-2(ab+bc+ca)<a^2+b^2+c^2\]
\[a^2+b^2+c^2<2(ab+bc+ca)\]
\[\Large \frac{a^2+b^2+c^2}{ab+bc+ca}<1\]
\[\LARGE 3 \lambda <2+2\]
\[\Huge \lambda< \frac{4}{3}\]

Answer 14

Its impossible on earth for me to think all that TBH

Answer 15

man thats soo cool !!!!

Answer 16

it looks neat

Answer 17

that property duh! cant imagine

Answer 18

that property is easy to visualize

Answer 19

sum of two sides must be greater than third side

Answer 20

we just need to digest above

Answer 21

i mean i got it but still its hard that it strikes me,+ all this too

Answer 22

same here, i dont have a clue until i saw the work u posted

Answer 23

HMM improve striking power dam me :D thanks for telling that property :p

Answer 24

haha comes wid practice id guess
that property is difference in lengths of sides, not simple subtraction

Answer 25

so, put absolute bars.
|a-b| < c

we're fine in above solution, as we are squaring

Answer 26

thanks! yes!

Answer 27

we dint use below information :
a not equal to b not equal to c

its useless