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  • Stuck here, help me understand: - Let , where. Then S represents : - Co-ordinate geometry - JEE Main

Let

S = \left \{ (x,y) \in \mathbb R^2: \frac{y^2}{1+r} - \frac{x^2}{1-r} = 1\right \},

where r\neq \pm 1. Then S represents :

  • Option 1)

    A hyperbola whose eccentricity is \frac{2}{\sqrt{1-r}} , when  0 < r <1

  • Option 2)

    A hyperbola whose eccentricity is \frac{2}{\sqrt{1+r}}  , when  0 < r <1

  • Option 3)

    An ellipse whose eccentricity is \frac{2}{\sqrt{1+r}} , when  r > 1

  • Option 4)

    An ellipse whose eccentricity is \frac{1}{\sqrt{1+r}}  , when  r > 1

Answers (1)

best_answer

 

Standard equation -

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

- wherein

a\rightarrow Semi major axis

b\rightarrow Semi minor axis

S=\frac{y^{2}}{1+r}-\frac{x^{2}}{1-r}

from the concept learnt

when r > 1

\frac{y^{2}}{1+r}-\frac{x^{2}}{1-r}=1

e=\sqrt{1-\frac{r-1}{r+1}}=\sqrt{\frac{r+1-r+1}{r+1}}=\sqrt{\frac{2}{r+1}}

correct option (3)


Option 1)

A hyperbola whose eccentricity is \frac{2}{\sqrt{1-r}} , when  0 < r <1

Option 2)

A hyperbola whose eccentricity is \frac{2}{\sqrt{1+r}}  , when  0 < r <1

Option 3)

An ellipse whose eccentricity is \frac{2}{\sqrt{1+r}} , when  r > 1

Option 4)

An ellipse whose eccentricity is \frac{1}{\sqrt{1+r}}  , when  r > 1

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