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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 = ?
6 years ago
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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). Answer: Let a = (x²+xy+y²) b= (y²+yz+z²) c= (z²+zx+x²) This equation reduces to proving that a^{3}+b^{3}+c^{3}=3abc This is possible if:
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).
Answer:
Let a = (x²+xy+y²)
b= (y²+yz+z²)
c= (z²+zx+x²)
This equation reduces to proving that a^{3}+b^{3}+c^{3}=3abc
This is possible if:
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)
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
=> a^{3}+b^{3}+c^{3}=3abc
which is
Hence proved.
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· a+b+c = 0
· or a=b=c
We realize by simple substitution that a+b+c is not 0 all the times
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