Multiple Choice Question (MCQ)

If a,x,y are real numbers such that a^2+(4+i)a+2xy+(x-y)i=0, then the trajectory of (x,y) represents a

  1. ×

    a line

  2. ×

    a circle centered at origin

  3. a circle centered not at origin

  4. ×

    an ellipse

Collected in the board: Imaginary number

Steven Zheng posted 1 week ago

Answer

  1. Given the equation

    a^2+(4+i)a+2xy+(x-y)i=0

    Rearrange by real and imaginary parts

    a^2+4a+2xy+( a+x-y)i=0

    To make the equation equal, we get the following two equations

    a^2+4a+2xy=0
    (1)

    and

    a+x-y=0
    (2)

    Then we get

    a =y-x
    (3)

    Substitute (3) to (1)

    (y-x)^2+4(y-x)+2xy=0

    Expand brackts and simplify

    x^2-4x+y^2+4y=0

    Add 8 on both sides and apply perfect square identity

    (x-2)^2+(y+2)^2 = 8

    which shows the trajectory of (x,y) represents a circle centered at point (2,-2).

    So C is the correct choice.

Steven Zheng posted 1 week ago

Scroll to Top