First slide

Hyperbola in conic sections

Question

$If the eccentricity of the hyperbola x^{2} - y^{2} \sec^{2} α = 5 is \sqrt{3} times the eccentricity of the ellipse x^{2} \sec^{2} α + y^{2} = 25, then a value of α is$

Moderate

A

$\frac{π}{6}$
B

$\frac{π}{4}$
C

$\frac{π}{3}$
D

$\frac{π}{2}$

Solution

For the hyperbola
$\frac{x^{2}}{5} - \frac{y^{2}}{5 \cos^{2} α} = 1$
we have
$e_{1}^{2} = 1 + \frac{b^{2}}{a^{2}} = 1 + \frac{5 \cos^{2} α}{5} = 1 + \cos^{2} α$
For the ellipse
$\frac{x^{2}}{25 \cos^{2} α} + \frac{y^{2}}{25} = 1$
we have
$e_{2}^{2} = 1 - \frac{25 \cos^{2} α}{25} = \sin^{2} α$
Given that
$\begin{array}{ll} e_{1} = \sqrt{3} e_{2} \\ ∴ e_{1}^{2} = 3 e_{2}^{2} \\ or 1 + \cos^{2} α = 3 \sin^{2} α \\ or 2 = 4 \sin^{2} α \\ or \sin α = \frac{1}{\sqrt{2}} \end{array}$

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