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KVPY Sample Paper KVPY Stream-SX Model Paper-29

  • question_answer
    Let \[S=\left\{ (x,y)\in {{R}^{2}}:\frac{{{y}^{2}}}{1+r}-\frac{{{x}^{2}}}{1-r}=1 \right\},\] where \[r\ne 1.\] Then S represents:

    A) a hyperbola whose eccentricity is\[\frac{2}{\sqrt{1-r}},\]when

    B) an ellipse whose eccentricity is\[\sqrt{\frac{2}{r+1}},\]when

    C) a hyperbola whose eccentricity is \[\frac{2}{\sqrt{r+1}},\] when

    D) an ellipse whose eccentricity is \[\frac{1}{\sqrt{r+1}},\] when

    Correct Answer: B

    Solution :

    \[\frac{{{y}^{2}}}{1+r}-\frac{{{x}^{2}}}{1-r}=1\]
    \[r>1\]\[\Rightarrow \]ellipse
    \[e=\sqrt{1-\left( \frac{r-1}{r+1} \right)}=\sqrt{\frac{2}{r+1}}\]


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