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If |z^2-1|=z^2+1,then z lies on what circle or straight line or parabola or an ellipse
Ans. Parabola.... since, |z^2-1|=z^2-1 , if z^2>1 =-(z^2-1) , if z^21 , z^2-1=z^2+1 or, 2=0 ; which can`t be possible.now, when -(z^2-1)=z^2+1 or, 2*z^2=0 or, z^2=0since , it is Parabola.
|z^2-1|=|z|^2+1_________________eq 1Let z= a+ib=> z^2= a^2-b^2+2abialso |z|= (a^2+b^2)^1/2putting the values in eq 1|(a^2-b^2-1)+2abi|= (a^2+b^2+1)[(a^2-b^2-1)^2+(2ab)^2]^1/2=(a^2+b^2+1)Squaring both sides and opening the squares We geta^4+b^4+1-2a^2b^2+2b^2-2a^2+4a^2b^2 = a^4+b^4+1+2a^2b^2+2b^2+2a^2 Cancelling Finally we get4a^2=0 Which suggests that it lies on an imaginary axis... Hope I solved your problem.I`d you find anything incorrect in my solution please let me know..
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