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Category: Algebra

If-the-imaginary-part-of-2z-1-iz-1-is-2-then-the-locus-of-the-point-representing-z-in-the-complex-plane-is-

Question Number 19734 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:\mathrm{the}\:\mathrm{imaginary}\:\mathrm{part}\:\mathrm{of}\:\frac{\mathrm{2}{z}\:+\:\mathrm{1}}{{iz}\:+\:\mathrm{1}}\:\mathrm{is}\:−\mathrm{2}, \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:\mathrm{representing} \\ $$$${z}\:\mathrm{in}\:\mathrm{the}\:\mathrm{complex}\:\mathrm{plane}\:\mathrm{is} \\ $$ Answered by ajfour last updated on 15/Aug/17 $$\:\:\frac{\mathrm{2z}+\mathrm{1}}{\mathrm{iz}+\mathrm{1}}=\frac{\left(\mathrm{2z}+\mathrm{1}\right)\left(\mathrm{1}−\mathrm{i}\bar {\mathrm{z}}\right)}{\left(\mathrm{iz}+\mathrm{1}\right)\left(\mathrm{1}−\mathrm{i}\bar…

The-locus-of-z-given-by-z-1-z-i-1-is-

Question Number 19732 by Tinkutara last updated on 15/Aug/17 $$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{given}\:\mathrm{by}\:\mid\frac{{z}\:−\:\mathrm{1}}{{z}\:−\:{i}}\mid\:=\:\mathrm{1}\:\mathrm{is} \\ $$ Answered by ajfour last updated on 15/Aug/17 $$\mathrm{z}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{perpendicular}\:\mathrm{bisector} \\ $$$$\mathrm{of}\:\mathrm{line}\:\mathrm{joining}\:\mathrm{z}=\mathrm{1}\:\mathrm{and}\:\mathrm{z}=\mathrm{i}\:\mathrm{that} \\ $$$$\:\:\mathrm{passes}\:\mathrm{through}\:\mathrm{origin}. \\…

If-z-1-2-z-1-then-the-locus-described-by-the-point-z-in-the-argand-diagram-is-a-

Question Number 19733 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:\mid{z}\:+\:\mathrm{1}\mid\:=\:\sqrt{\mathrm{2}}\mid{z}\:−\:\mathrm{1}\mid,\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{described}\:\mathrm{by}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{in}\:\mathrm{the}\:\mathrm{argand} \\ $$$$\mathrm{diagram}\:\mathrm{is}\:\mathrm{a} \\ $$ Answered by ajfour last updated on 15/Aug/17 $$\mathrm{let}\:\mathrm{z}=\mathrm{x}+\mathrm{iy} \\…

If-z-x-iy-and-z-2i-1-then-1-z-lies-on-x-axis-2-z-lies-on-y-axis-3-z-lies-on-a-circle-4-None-of-these-

Question Number 19730 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{and}\:\mid{z}\:−\:\mathrm{2}{i}\mid\:=\:\mathrm{1},\:\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:{x}-\mathrm{axis} \\ $$$$\left(\mathrm{2}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:{y}-\mathrm{axis} \\ $$$$\left(\mathrm{3}\right)\:{z}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{None}\:\mathrm{of}\:\mathrm{these} \\ $$ Answered by ajfour last updated…

x-y-N-18-x-2-y-2-2-9-xy-2592-find-xy-

Question Number 150797 by mathdanisur last updated on 15/Aug/21 $$\mathrm{x};\mathrm{y}\in\mathbb{N} \\ $$$$\frac{\mathrm{18}^{\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:+\:\boldsymbol{\mathrm{y}}^{\mathrm{2}} }{\mathrm{2}}} }{\mathrm{9}^{\boldsymbol{\mathrm{xy}}} }\:=\:\mathrm{2592}\:\:\Rightarrow\:\mathrm{find}\:\:\boldsymbol{\mathrm{xy}}=? \\ $$ Answered by nimnim last updated on 15/Aug/21…

if-y-z-x-10-3-and-x-z-3-4-find-x-y-

Question Number 150792 by mathdanisur last updated on 15/Aug/21 $$\mathrm{if}\:\:\:\frac{\mathrm{y}+\mathrm{z}}{\mathrm{x}}\:=\:\frac{\mathrm{10}}{\mathrm{3}}\:\:\mathrm{and}\:\:\frac{\mathrm{x}}{\mathrm{z}}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{find}\:\:\frac{\mathrm{x}}{\mathrm{y}}\:=\:? \\ $$ Answered by nimnim last updated on 15/Aug/21 $${y}=\frac{\mathrm{10}{x}−\mathrm{3}{z}}{\mathrm{3}}\:\:\:\:{and}\:\:\:\:\mathrm{3}{z}=\mathrm{4}{x} \\ $$$$\therefore\frac{{x}}{{y}}=\frac{{x}}{\frac{\mathrm{10}{x}−\mathrm{4}{x}}{\mathrm{3}}}\:=\frac{\mathrm{1}}{\mathrm{2}} \\…