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`        a  vertical line divides the triangle with vertices (0,0),(1,1),(9,1) in  xy plane into 2 regions of equal area.the equation of line is x = ?`
7 years ago

105 Points
```										If x+ y+ z=0
Prove that:
(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).
Let a = (x²+xy+y²)
b= (y²+yz+z²)
c= (z²+zx+x²)
This equation reduces to proving that a3+b3+c3=3abc
This is possible if:

a+b+c = 0
or a=b=c

We realize by simple substitution(like taking x,y,z = (-1,0,1),(-2,0,2)) that a+b+c is not 0 all the times
Now considering a=b
If and only if
(x²+xy+y²) = (y²+yz+z²)
If and only if
x²+xy = yz+z²
i.e.,        x(x+y)=z(y+z)
Using the fact that x+y+z=0, we see this is nothing but,
x(-z)=z(-x)
Hence a=b=c
=> a3+b3+c3=3abc
which is
(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).
Hence proved.
Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.

If x+ y+ z=0
Prove that:
(x²+xy+y²)³+ (y²+yz+z²)³+ (z²+zx+x²)³ = 3(x²+xy+y²). (y²+yz+z²) (z²+zx+x²).
Let a = (x²+xy+y²)
b= (y²+yz+z²)
c= (z²+zx+x²)
This equation reduces to proving that a3+b3+c3=3abc
This is possible if:
·         a+b+c = 0
·         or a=b=c
We realize by simple substitution that a+b+c is not 0 all the times
Now considering a=b
If and only if
(x²+xy+y²) = (y²+yz+z²)
If and only if
x²+xy = yz+z²
i.e.,        x(x+y)=z(y+z)
Using the fact that x+y+z=0, we see this is nothing but,
x(-z)=z(-x)
Hence proved.

```
7 years ago
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