InterviewSolution
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InterviewSolution
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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. 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`. |
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Answer» 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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