Results 1 to 4 of 4

Math Help - Complex numbers

  1. #1
    Member
    Joined
    Sep 2006
    Posts
    77

    Complex numbers

    1. If z=x+iy satisfies \mid z-4 \mid + \mid z+4 \mid =10 show that

      (x/5)^2+(y/3)^2=1


    2. Conversely, if x and y satisfy (x/5)^2+(y/3)^2=1, then show that z=x+iy satisfies \mid z-4 \mid + \mid z+4 \mid =10.



    Not a clue how to do this, nothing seems to work.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,658
    Thanks
    1615
    Awards
    1
    These may help you. You can square both sides.
    \left| {z - 4} \right|^2  = \left( {z - 4} \right)\left( {\overline z  - 4} \right) = z\overline z  - 4z - 4\overline z  + 16

    \left| {z + 4} \right|^2  = \left( {z + 4} \right)\left( {\overline z  + 4} \right) = z\overline z  + 4z + 4\overline z  + 16

    \left| {z - 4} \right|^2  + \left| {z + 4} \right|^2  = 2z\overline z  + 32

    \left( {z + 4} \right)\left( {z - 4} \right) = z^2  - 16
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,738
    Thanks
    645
    Hello, Thomas!

    Another approach . . .


    If z\:=\:x+iy satisfies |z-4| + |z + 4|\:=\:10

    show that: . \left(\frac{x}{5}\right)^2 + \left(\frac{y}{3}\right)^2\:=\:1
    |z-4| is the distance from z to the point (4,0).

    |z+4| is the distance from z to the point (-4,0).

    If the sum of the distances to two fixed points is a constant,
    . . the locus of z is an ellipse, whose equation can be derived
    . . with the Distance Formula (and a lot of algebra).

    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Sep 2006
    Posts
    77
    Thanks. I can do (1) by taking \left| z+4 \right| to be \sqrt{(x+4)^2+y^2} and the same for the other. However I'm having trouble doing the opposite, starting with the ellipse eqn and finding that z=x+iy...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. raising complex numbers to complex exponents?
    Posted in the Advanced Math Topics Forum
    Replies: 10
    Last Post: March 25th 2011, 10:02 PM
  2. Replies: 1
    Last Post: September 27th 2010, 03:14 PM
  3. Replies: 2
    Last Post: February 7th 2009, 06:12 PM
  4. Replies: 1
    Last Post: May 24th 2007, 03:49 AM
  5. Complex Numbers- Imaginary numbers
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 24th 2007, 12:34 AM

Search Tags


/mathhelpforum @mathhelpforum