For the Hyperbola   \frac{x^{2}}{\cos ^{2}\alpha }-\frac{y^{2}}{\sin ^{2}\alpha }=1,  which of the following remains constant when \alpha varies ?

  • Option 1)

    abscissae of vertices

  • Option 2)

    abscissae of foci

  • Option 3)

    eccentricity

  • Option 4)

    directrix

 

Answers (1)

As we learnt in 

Coordinates of foci -

\pm ae,o

- wherein

For the ellipse  

\frac{x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}= 1

 

 \frac{x^{2}}{cos^{2}\alpha}-\frac{y^{2}}{sin^{2}\alpha}=1

Here a=cos\alpha ,b=sin\alpha

e=\sqrt{1+\frac{sin^2\alpha }{cos^2\alpha }} =sec\alpha

Coordnita of focus = ae

=cos\alpha \times sec\alpha =1


Option 1)

abscissae of vertices

Incorrect

Option 2)

abscissae of foci

Correct

Option 3)

eccentricity

Incorrect

Option 4)

directrix

Incorrect

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