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

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

Question
Conic sections
asked 2021-01-07
Find the vertices, foci, directrices, and eccentricity of the curve wich polar conic section Consider the equation \(r^{2} = \sec\ 2\ \theta\)

Answers (1)

2021-01-08
On comparing the equation with the equation \(\frac{3^{2}}{a^{2}}\ -\ \frac{y^{2}}{b^{2}}=1\) We have \(a = b = 1\) also \(c^{2} = a^{2}\ +\ b^{2} = 1\ +\ 1 = 2\)
\(c = \sqrt{2}\) The equation of hyperbola is given by \(\frac{x^{2}}{a^{2}}\ -\ \frac{y^{2}}{b^{2}}=1\) Vertex is \((\pm\ a\ 0)\ \rightarrow\ (\pm\ 1,\ 0)\) Foci is \((\pm\ c,\ 0)\ \rightarrow (\pm\ \sqrt{2},\ 0)\) Directrix is \(x =\ \pm\ \frac{a^{2}}{c}\ =\ \pm\ \frac{1}{\sqrt{2}}\) Eccentricity \(e = \frac{c}{a} = \sqrt{2}\)
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