Cheyanne Leigh

Answered

2021-01-07

Find the vertices, foci, directrices, and eccentricity of the curve wich polar conic section Consider the equation ${r}^{2}=\mathrm{sec}\text{}2\text{}\theta $

Answer & Explanation

Bertha Stark

Expert

2021-01-08Added 96 answers

On comparing the equation with the equation $\frac{{3}^{2}}{{a}^{2}}\text{}-\text{}\frac{{y}^{2}}{{b}^{2}}=1$
We have $a=b=1$ also
${c}^{2}={a}^{2}\text{}+\text{}{b}^{2}=1\text{}+\text{}1=2$

$c=\sqrt{2}$
The equation of hyperbola is given by $\frac{{x}^{2}}{{a}^{2}}\text{}-\text{}\frac{{y}^{2}}{{b}^{2}}=1$
Vertex is $(\pm \text{}a\text{}0)\text{}\to \text{}(\pm \text{}1,\text{}0)$
Foci is $(\pm \text{}c,\text{}0)\text{}\to (\pm \text{}\sqrt{2},\text{}0)$
Directrix is $x=\text{}\pm \text{}\frac{{a}^{2}}{c}\text{}=\text{}\pm \text{}\frac{1}{\sqrt{2}}$
Eccentricity $e=\frac{c}{a}=\sqrt{2}$

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