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Let the complex number z=x+iy be such that 2z+i2z−3i is purely imaginary. If x+y2=0, then y4+y2−y is equal to:
Explanation
2z+i2z−3i=2x+2iy+i2x+2iy−3i
2x+i(2y+1)2x+i(2y−3)×2x−i(2y+1)2x−i(2y+1)
2z+i2z−3i=(4x2+(2y+1)24x2+(2y−3)(2y+1))+i(4x2+(2y+1)22x(2y−3)−2x(2y+1))
Now 2z+i2z−3i is purely imaginary if:
4x2+(2y+1)24x2+(2y−3)(2y+1)=0
but x+y2=0 is given
so x=−y2 and x2=y4
Explanation
2z+i2z−3i=2x+2iy+i2x+2iy−3i
2x+i(2y+1)2x+i(2y−3)×2x−i(2y+1)2x−i(2y+1)
2z+i2z−3i=(4x2+(2y+1)24x2+(2y−3)(2y+1))+i(4x2+(2y+1)22x(2y−3)−2x(2y+1))
Now 2z+i2z−3i is purely imaginary if:
4x2+(2y+1)24x2+(2y−3)(2y+1)=0
but x+y2=0 is given
so x=−y2 and x2=y4
