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Question

For the hyperbola x2cos2αy2sin2α=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,

x2cos2y2sin2=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(e21)

e21=b2a2

=sin2cos2=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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