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

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Uniteasy Admin · Accepted Answer

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
$$ a^2+4a+2xy=0$$n
and 
$$a+x-y=0$$n
Then we get 
$$a =y-x$$n
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.

Multiple Choice Question (MCQ)

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