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Question

# For the hyperbola x2cos2α−y2sin2α=1 Which one of the following remain constant with change of α ?

A

Abscissa of vertices

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B

abscissa of foci

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C

Eccentricity

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D

directrix

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Solution

## The correct option is B abscissa of foci Given hyperbola is, x2cos2∝−y2sin2∝=1 Comparing with a standard hyperbola we get, a2=cos2α b2=sin2α We are asked to find which of the given options remain unchanged with change in α. For finding that, we need to see which of the quantities remain independent of α. b2=a2(e2−1) e2−1=b2a2 =sin2∝cos2∝=tan2∝ e=|sec ∝| directrix,x=±ae=±cos ∝sec ∝ Abscissa of focus=±ae =±cosα|sec ∝| =±1 Which is independent of α. Hence answer (b).

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