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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 = ?

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 = ?

Grade:11

1 Answers

suryakanth AskiitiansExpert-IITB
105 Points
13 years ago

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 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.

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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 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.

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