Home
Class 11
MATHS
If (x+i y)^3=u+i v ,then show that u/x+...

If `(x+i y)^3=u+i v ,`then show that `u/x+v/y=4(x^2-y^2)`.

Text Solution

Verified by Experts

The correct Answer is:
N/a
`" "(x+iy)^(3)=u+iv`
`rArr" "x^(3)+ 3x^(2) iy+ 3x i^(2)y^(2)+i^(3)y^(3)= u+iv`
`rArr" "x+3x^(2)iy-3 xy^(2) -iy^(3) =u+iv`
Equate real and imaginary parts, we get
`" "x^( 3)- 3xy^(2)=u and 3x^(2)y-y^(3)=upsilon`
`rArr" "(u)/(x)=x^(2) -3y^(2) and (upsilon)/(y)=3x^(2)-y^(2)`
Now, ` (u)/( x)+( upsilon)/(y)=x^(2)- 3y^(2)+ 3x^(2)-y^(2)`
`" "=4x^(2)- 4y^(2) `
`" "=4(x^(2)-y^(2))`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS AND QUADRATIC EQUATION

    NAGEEN PRAKASHAN ENGLISH|Exercise EXERCISE 5.3|10 Videos

  • BINOMIAL THEOREM

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exericse|20 Videos

  • CONIC SECTION

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|8 Videos

Similar Questions

Explore conceptually related problems

If (x+yi)^(3)=u + vi , prove that (u)/(x) +(v)/(y) = 4(x^(2)-y^(2))

If 3x + y i= x + 2y , then 2x - y =

If x+iy=3/(2+costheta +i sin theta) , then show that x^2+y^2=4x-3

If (x+i y)^(1//3)=a+i b ,x ,y ,a b in Rdot Show that x/a+y/b=4(a^2-b^2) (ii) x/a-y/b=2(a^2+b^2)

If (5x + 6y)/(5u + 6v) = (5x - 6y)/(5u - 6v) : then prove that x : y = u : v .

If z= x + yi and |2z + 1| = |z- 2i| , show that 3(x^(2) + y^(2)) + 4(x-y)=3

If (x , y) and (x ,y) are the coordinates of the same point referred to two sets of rectangular axes with the same origin and it u x+v y , where u and v are independent of xa n dy , becomes V X+U Y , show that u^2+v^2=U^2+V^2dot

If (x+i y)(p+i q)=(x^2+y^2)i , prove that x=q ,y=pdot

If x+ yi= (u+ vi)/(u-yi) , prove that x^(2) + y^(2)=1

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v,v),(u'',v'',2v'):}| .

NAGEEN PRAKASHAN ENGLISH-COMPLEX NUMBERS AND QUADRATIC EQUATION -MISCELLANEOUS EXERCISE
  1. Evaluate : [i^(18)+(1/i)^(25)]^3

    Text Solution

    |

  2. For any two complex numbers z1and z2, prove that R e(z1z2)=R ez1R e z2...

    Text Solution

    |

  3. Reduce (1/(1-4i)-2/(1+i))((3-4i)/(5+i))to the standard form.

    Text Solution

    |

  4. If under root of (a+i b)/(c+i d)=x+i y , Prove (a^2+b^2)/(c^2+d^2)=(x...

    Text Solution

    |

  5. Convert the following in the polar form : (i) (1+7i)/((2-i)^2) (ii) (...

    Text Solution

    |

  6. Solve the equation : 3x^2-4x+(20)/3=0

    Text Solution

    |

  7. Solve the equation :x^2-2x+3/2=0

    Text Solution

    |

  8. Solve the equation :27 x^2-10 x+1=0

    Text Solution

    |

  9. Solve the following quadratic: 21 x^2-28 x+10=0

    Text Solution

    |

  10. If z1=2-i ,z2=1+i ,find |(z1+z2+1)/(z1-z2+i)|

    Text Solution

    |

  11. If a + i b =((x+i)^2)/(2x^2+1),prove that a^2+b^2=((x^2+1)^2)/((2x^2+...

    Text Solution

    |

  12. If z1=2-i ,\ +2=-2+i , find : R e((z1z2)/(z1))

    Text Solution

    |

  13. Find the modulus and argument of the complex number (1+2i)/(1-3i).

    Text Solution

    |

  14. Find the real numbers x and y if (x-i y)(3+5i)is the conjugate of -6-...

    Text Solution

    |

  15. Find the modulus of (1+i)/(1-i)-(1-i)/(1+i)

    Text Solution

    |

  16. If (x+i y)^3=u+i v ,then show that u/x+v/y=4(x^2-y^2).

    Text Solution

    |

  17. If alphaand betaare different complex numbers with |beta|=1,then fin...

    Text Solution

    |

  18. Find the number of non-zero integral solution of the equation |1-i|^x=...

    Text Solution

    |

  19. If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a^2 +...

    Text Solution

    |

  20. If ((1+i)/(1-i))^m=1, then find the least positive integral value of m...

    Text Solution

    |