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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 +- 1`, Then S represents

A

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

B

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

C

an ellipse whose eccentricity is `sqrt((2)/(r+1)) " when "r gt 1`.

D

an ellipse whose eccentricity is `(1)/(sqrt(r+1)) " when "r gt 1`.

Text Solution

Verified by Experts

Given, `S={(x,y) in R^(2): (y^(2))/(1+r)-(x^(2))/(1-r)=1}`
`={(x,y) in R^(2): (y^(2))/(1+r)+(x^(2))/(r-1)=1}`
For `r gt 1,(y^(2))/(1+r)+(x^(2))/(r-1)=1`, represents a vertical ellipse.
`" "[ because " for " r gt 1, r-1 lt r+1 and r-1 gt 0]`
Now, eccentricity (e)`=sqrt(1-(r-1)/(r+1))`
`[ because " For " (x^(2))/(a^(2))-(y^(2))/(b^(2))=1, a lt b, e -sqrt(1-(a^(2))/(b^(2)))]`
`=sqrt(((r+1)-(r-1))/(r+1))`
`=sqrt((2)/(r+1))`
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