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Let S={(x,y) in R^(2):(y^(2))/(1+r)-(x^(...

Let `S={(x,y) in R^(2):(y^(2))/(1+r)-(x^(2))/(1-r)=1}`, where `r ne pm 1`. Then S represents:

A

An ellipse whose eccentricity is `(1)/(sqrt(r+1))`, where `r gt 1`

B

A hyperbola whose eccentricity is `(2)/(sqrt(1-r))`, when `0 lt r lt 1`.

C

An ellipse whose eccentricity is `sqrt((2)/(r+1))`, when `r gt 1`

D

A hyperbola whose eccentricity is `(2)/(sqrt(r+1))`, where `0 lt r lt 1`.

Text Solution

Verified by Experts

The correct Answer is:
C
`(y^(2))/(1+r)-(x^(2))/(1-r)=1`
Ellipse : `(x^(2))/(r-1)+(y^(2))/(r+1)=1 " when " r gt 1`
`e= sqrt( 1-(r-1)/(r+1))= sqrt((2)/(r+1))`
Hyperbola : `(x^(2))/(1-r)-(y^(2))/(1+r)=-1 " when " 0 lt r lt 1`
`e=sqrt(1+(1-r)/(1+r))=sqrt((2)/(r+1))`
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