For the hyperbola x2-cos 2-y2-sin 2-1- which of the following remains fixed when - varies-A- directrixB- verticesC- fociD- eccentricity

The correct option is C fociFoci of hyperbola are (± ae,0) = [ ±√(a^2+b^2),0 ] = [ ±√(cos^2α+sin^2α),0 ] nbsp; =(± 1,0) Since, foci are independent of α. ⇒ ...

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Standard XII

Mathematics

Eccentric Angle : Ellipse

For the hyper...

Question

For the hyperbola $\frac{x^{2}}{{cos}^{2} α} - \frac{y^{2}}{{sin}^{2} α} = 1$ , which of the following remains fixed when $α$ varies?

directrix

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vertices

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foci

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eccentricity

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Solution

The correct option is C foci
Foci of hyperbola are $(\pm a e, 0)$
$= [\pm \sqrt{a^{2} + b^{2}}, 0]$
$= [\pm \sqrt{{cos}^{2} α + {sin}^{2} α}, 0]$
$= (\pm 1, 0)$

Since, foci are independent of $α .$
$\Rightarrow$ Foci are fixed.

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For the hyperbola x2cos2α−y2sin2α=1, which of the following remains fixed when α varies?

The correct option is C fociFoci of hyperbola are (±ae,0) =[±√a2+b2,0] =[±√cos2α+sin2α,0] =(±1,0) Since, foci are independent of α. ⇒ Foci are fixed.

For the hyperbola $\frac{x^{2}}{{cos}^{2} α} - \frac{y^{2}}{{sin}^{2} α} = 1$ , which of the following remains fixed when $α$ varies?

The correct option is C foci
Foci of hyperbola are $(\pm a e, 0)$
$= [\pm \sqrt{a^{2} + b^{2}}, 0]$
$= [\pm \sqrt{{cos}^{2} α + {sin}^{2} α}, 0]$
$= (\pm 1, 0)$

Since, foci are independent of $α .$
$\Rightarrow$ Foci are fixed.