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  4. Possible ordered pair(s) ( x , y ) on the complex plane satisfying the relation (4-3 i) x2+(3+2 i) x y=4 y2-(x2/2)+(3 x y-2 y2) i is/are

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Q. Possible ordered pair(s) $( x , y )$ on the complex plane satisfying the relation $(4-3 i) x^2+(3+2 i) x y=4 y^2-\frac{x^2}{2}+\left(3 x y-2 y^2\right) i$ is/are

Complex Numbers and Quadratic Equations

Solution:

We have $\left(\frac{9 x^2}{2}+3 x y-4 y^2\right)+\left(-3 x^2+2 x y-3 x y+2 y^2\right) i=0+0 i$ Hence $9 x^2+6 x y-8 y^2=0 \Rightarrow (3 x+4 y)(3 x-2 y)=0 \ldots(1)$ and $ 3 x^2+x y-2 y^2=0 \Rightarrow (3 x-2 y)(x+y)=0 \ldots .(2)$ From (1) and (2), we get $3 x=2 y \Rightarrow (C)$ and (D)