Question

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

Conic sections
Find the vertices, foci, directrices, and eccentricity of the curve wich polar conic section Consider the equation $$r^{2} = \sec\ 2\ \theta$$
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 $$e = \frac{c}{a} = \sqrt{2}$$