let z=x+iy be the complex number where x and y are integers .then what is the area of rectangle whose vertices are the roots of the equation Z(z)^3+z(Z)^3=350(where Z represents the conjugate of z)

Dear Pratish,

Let Z= Complex no.=x + iy.

Let Z= Z bar=x-iy.

 Now,    ZZ³ + ZZ³  = 352

         » ZZ.Z²   +  ZZ.Z²  = 350

         » ZZ (Z²   +  Z² )  = 350

         » (x+iy).(x-iy) [ (x-iy)² + (x+iy)² ] =350

         » (x² + y²).(2x² - 2y²) =350

         » (x² + y²).(x² - y²) =175

         » 25.7=175

         » x² + y² = 25  &  x² - y² = 7 

         »x²=16 & y²=9

         »x=+4 or -4     &      y=+3 or-3

         »So 4 pts of rectangle are  (4,3) , (4,-3) , (-4,3) , (-4,-3).

         »Hence, length of rectangle=8 units       &      breadth of rectangle=6 units.

         »So, area of rectangle=length . breadth =8.6=48 sq. units.

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Approved

DEAR PRATISH!

Your ANSWER IS

let z = x+iy

and

    z = x-iy

therefore,

→ zz³ + zz³ = 350

or,zz(z² + z²) = 30

or,|z|² [(x+iy)²+(x-iy)²] = 350                (if z=x+iy)

or,(x²+y²)[x²-y²] = 350/2 = 175 = 25*7   (by analogy)

or,(x²+y²)=25 & (x²-y²)=7

        or, x=+4 or -4     &      y=+3 or-3

or,So 4 pts of rectangle are  (4,3) , (4,-3) , (-4,3) , (-4,-3).

         or,Hence, length of rectangle=8 units       &      breadth of rectangle=6 units.

         or,So, area of rectangle=length . breadth =8.6=48 sq. units. ANSWER

Approved

coordinates are (4,3)      (-4,3)     (-4,-3)        

(4,-3)        area is    48 units

dear student,

Let Z= Complex no.=x + iy.
Let Z= Z bar=x-iy.
 Now,    ZZ3 + ZZ3  = 352
         » ZZ.Z2   +  ZZ.Z2  = 350
         » ZZ (Z2   +  Z2 )  = 350
         » (x+iy).(x-iy) [ (x-iy)2 + (x+iy)2 ] =350
         » (x2 + y2).(2x2 - 2y2) =350
         » (x2 + y2).(x2 - y2) =175          » 25.7=175
         » x2 + y2 = 25  &  x2 - y2 = 7 
         »x2=16 & y2=9
         »x=+4 or -4     &      y=+3 or-3
         »So 4 pts of rectangle are  (4,3) , (4,-3) , (-4,3) , (-4,-3).
         »Hence, length of rectangle=8 units       &      breadth of rectangle=6 units.
         »So, area of rectangle=length . breadth =8.6=48 sq. units.

hope you're now clear with the problem now, and don't ever try to challenge the faculty, you should know to respect them