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Category: Coordinate Geometry

R-r-2-R-2-R-r-2-R-2-r-2-2rR-R-2-R-2-r-2-2rR-R-2-4rR-r-R-4-1cm-A-S-pir-2-pi-1cm-2-picm-2-

Question Number 150793 by cherokeesay last updated on 15/Aug/21 $$\left({R}−{r}\right)^{\mathrm{2}} \:+\:{R}^{\mathrm{2}} \:=\:\left({R}\:+\:{r}\right)^{\mathrm{2}} \:\Leftrightarrow \\ $$$${R}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} −\mathrm{2}{rR}\:+\:{R}^{\mathrm{2}} \:=\:{R}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} \:+\mathrm{2}{rR}\:\Rightarrow \\ $$$${R}^{\mathrm{2}} \:=\:\mathrm{4}{rR}\:\Rightarrow\:{r}\:=\:\frac{{R}}{\mathrm{4}}\:=\:\mathrm{1}{cm} \\ $$$$\mathscr{A}_{{S}}…

A-point-P-x-y-moves-such-that-its-perpendicular-distance-from-the-line-12x-5y-1-0-is-always-3-units-Find-the-equation-that-describes-the-locus-precisely-

Question Number 150738 by nadovic last updated on 15/Aug/21 $$\mathrm{A}\:\mathrm{point}\:\mathrm{P}\left({x},\:{y}\right)\:\mathrm{moves}\:\mathrm{such}\:\mathrm{that}\:\mathrm{its} \\ $$$$\mathrm{perpendicular}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{12}{x}\:+\:\mathrm{5}{y}\:−\:\mathrm{1}\:=\:\:\mathrm{0}\:\mathrm{is}\:\mathrm{always}\:\mathrm{3}\:{units}.\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{that}\:\mathrm{describes}\:\mathrm{the}\: \\ $$$$\mathrm{locus}\:\mathrm{precisely}. \\ $$ Commented by nadovic last updated…

Question-150596

Question Number 150596 by mr W last updated on 14/Aug/21 Commented by mr W last updated on 14/Aug/21 $${a}\:{more}\:{challenging}\:{case}:\:\mathrm{3}{D}\:{case} \\ $$$${three}\:{vertex}\:{of}\:{a}\:{tetrahedron}\:{lie}\:{on} \\ $$$${the}\:{coordinate}\:{axes}.\:{the}\:{fourth}\:{one} \\ $$$${lies}\:{on}\:{the}\:{sphere}\:{with}\:{radius}\:{R}\:{and}…

Question-19021

Question Number 19021 by ajfour last updated on 03/Aug/17 Commented by ajfour last updated on 03/Aug/17 $$\mathrm{If}\:\:\:\:\:\:\phi=\mathrm{tan}^{−\mathrm{1}} \left[\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)\mathrm{cot}\:\theta\right]+\theta \\ $$$$\Rightarrow\:\mathrm{tan}\:\left(\phi−\theta\right)=\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)\mathrm{cot}\:\theta \\ $$$$\:\:\:\frac{\mathrm{tan}\:\phi−\mathrm{tan}\:\theta}{\mathrm{1}+\mathrm{tan}\:\phi\mathrm{tan}\:\theta}=\frac{\mathrm{cot}\:\theta}{\:\sqrt{\mathrm{2}}+\mathrm{1}} \\ $$$$\Rightarrow\:\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\:\left(\mathrm{tan}\:\phi−\mathrm{tan}\:\theta\right) \\…

find-for-equation-of-image-ellipse-x-2-9-y-2-8-1-if-reflected-with-line-x-y-4-

Question Number 84531 by jagoll last updated on 14/Mar/20 $$\mathrm{find}\:\mathrm{for}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{image}\:\mathrm{ellipse} \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{9}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{8}}\:=\:\mathrm{1}\:\mathrm{if}\:\mathrm{reflected}\:\mathrm{with}\:\mathrm{line} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:=\:−\mathrm{4} \\ $$ Commented by jagoll last updated on 14/Mar/20…