For hyperbola x 2 cos 2 - y 2 sin 2 = 1 , which of the following remains constant with change in [2003]

Question

For hyperbola $\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1,$ which of the following remains constant with change in $' α'$ [2003]

Smart Achievers · Accepted Answer

(2)

$Given equation of hyperbola is$

$\frac{x^{2}}{\cos^{2} α} - \frac{y^{2}}{\sin^{2} α} = 1$ $\Rightarrow a = \cos α, b = \sin α$

$\Rightarrow e = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \sqrt{1 + \tan^{2} α} = \sec α$

$\Rightarrow a e = \cos α \cdot \sec α = 1$ $∴$ $Foci (\pm 1, 0)$

$Hence, foci remain constant with respect to α .$

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