Question Number 19738 by Tinkutara last updated on 15/Aug/17 $$\mathrm{Locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid{iz}\:−\:\mathrm{1}\mid\:+\:\mid{z}\:−\:{i}\mid\:=\:\mathrm{2}\:\mathrm{is} \\ $$ Commented by math khazana by abdo last updated on 22/Jun/18 $${let}\:{z}\:={x}+{iy}\:\:\left({e}\right)\:\Leftrightarrow\mid{ix}−{y}−\mathrm{1}\mid\:+\mid{x}+{i}\left({y}−\mathrm{1}\right)\mid=\mathrm{2}…
Question Number 150804 by mathdanisur last updated on 15/Aug/21 Answered by aleks041103 last updated on 15/Aug/21 $${Let}\:{P}=\left({b}−{a}\right)\left({c}−{a}\right)\left({d}−{a}\right)\left({c}−{b}\right)\left({d}−{b}\right)\left({d}−{c}\right). \\ $$$${Obviously}\:{P}\:\:{is}\:{the}\:{product}\:{of}\:{all} \\ $$$${possible}\:{differences}\:{between}\:{a},{b},{c},{d}. \\ $$$$\mathrm{1}.\:{Divisibility}\:{by}\:\mathrm{3},\:{i}.{e}.\:\mathrm{3}\mid{P}. \\ $$$${If}\:{any}\:{two}\:{of}\:{the}\:{four}\:{numbers}…
Question Number 19735 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{z}\:=\:\lambda\:+\:\mathrm{3}\:+\:{i}\sqrt{\mathrm{5}\:−\:\lambda^{\mathrm{2}} },\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a} \\ $$ Answered by ajfour last updated on 15/Aug/17 $$\mathrm{if}\:\mathrm{we}\:\mathrm{let}\:\mathrm{z}=\mathrm{x}+\mathrm{iy} \\ $$$$\left(\mathrm{x}−\lambda−\mathrm{3}\right)+\mathrm{i}\left(\mathrm{y}−\sqrt{\mathrm{5}−\lambda^{\mathrm{2}}…
Question Number 150807 by mathdanisur last updated on 15/Aug/21 $$\mathrm{For}\:\mathrm{matris}\:\mathrm{solution}: \\ $$$$\begin{cases}{\mathrm{2x}\:-\:\mathrm{3y}\:=\:\mathrm{8}}\\{\mathrm{x}\:+\:\mathrm{5y}\:=\:-\:\mathrm{9}}\end{cases} \\ $$ Answered by maged last updated on 15/Aug/21 $$\Delta=\mid\underbrace{ } \\ $$…
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 $$\:\:\frac{\mathrm{2z}+\mathrm{1}}{\mathrm{iz}+\mathrm{1}}=\frac{\left(\mathrm{2z}+\mathrm{1}\right)\left(\mathrm{1}−\mathrm{i}\bar {\mathrm{z}}\right)}{\left(\mathrm{iz}+\mathrm{1}\right)\left(\mathrm{1}−\mathrm{i}\bar…
Question Number 19732 by Tinkutara last updated on 15/Aug/17 $$\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} \\ $$$$\mathrm{of}\:\mathrm{line}\:\mathrm{joining}\:\mathrm{z}=\mathrm{1}\:\mathrm{and}\:\mathrm{z}=\mathrm{i}\:\mathrm{that} \\ $$$$\:\:\mathrm{passes}\:\mathrm{through}\:\mathrm{origin}. \\…
Question Number 19733 by Tinkutara last updated on 15/Aug/17 $$\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 $$\mathrm{let}\:\mathrm{z}=\mathrm{x}+\mathrm{iy} \\…
Question Number 19729 by Joel577 last updated on 15/Aug/17 $$\mathrm{2}{x}\:+\:\mathrm{9}{y}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:−\:\mathrm{45}{y}^{\mathrm{2}} \:+\:{xy}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{xy} \\ $$ Answered by mrW1 last updated on…
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…
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…