Babenzgs

2022-01-25

Separate equations of the pair of lines ${x}^{2}+2xy\mathrm{sec}\theta +{y}^{2}=0$

caoireoilns

Expert

${x}^{2}+2xy\mathrm{sec}\theta +{y}^{2}=0$
$\frac{{x}^{2}}{{y}^{2}}+2\frac{x}{y}\mathrm{sec}\theta +1=0$
$\frac{x}{y}=\frac{-2\mathrm{sec}\theta ±\sqrt{4{\mathrm{sec}}^{2}\theta -4}}{2}=-\mathrm{sec}\theta ±\mathrm{tan}\theta =\frac{-1±\mathrm{sin}\theta }{\mathrm{cos}\theta }$
$\frac{-1±\mathrm{sin}\theta }{\mathrm{cos}\theta }=\frac{-1±\mathrm{sin}\theta }{\mathrm{cos}\theta }\cdot \frac{-1\mp \mathrm{sin}\theta }{-1\mp \mathrm{sin}\theta }$
$=\frac{1-{\mathrm{sin}}^{2}\theta }{\mathrm{cos}\theta \left(-1\mp \mathrm{sin}\theta \right)}$
$=\frac{{\mathrm{cos}}^{2}\theta }{\mathrm{cos}\theta \left(-1\mp \mathrm{sin}\theta \right)}$
$=\frac{\mathrm{cos}\theta }{-1\mp \mathrm{sin}\theta }$

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