JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer If \[(x+iy)(p+iq)=({{x}^{2}}+{{y}^{2}})i\], then

    A) \[p=x,q=y\]

    B) \[p={{x}^{2}},\,\,q={{y}^{2}}\]

    C) \[x=q,y=p\]

    D) None of these

    Correct Answer: C

    Solution :

    \[(x+iy)(p+iq)=({{x}^{2}}+{{y}^{2}})i\] Þ   \[(xp-yq)+i(xq+yp)=({{x}^{2}}+{{y}^{2}})i\] Þ   \[xp-yq=0,xq+yp={{x}^{2}}+{{y}^{2}}\] Þ  \[\frac{x}{q}=\frac{y}{p}\]and\[xq+yp={{x}^{2}}+{{y}^{2}}\] Let \[\frac{x}{q}=\frac{y}{p}=\lambda \]. then \[x=\lambda q,y=\lambda p\] \\[xq+yp={{x}^{2}}+{{y}^{2}}\]Þ \[\lambda ={{\lambda }^{2}}\]Þ \[\lambda =1\] \ \[x=q,y=p\].

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