QJEEM19Mafe38: JEE Main 2019 Mathematics - Class 11 JEEM19Mafe38 Solution: Let S = Then S represents :

Question

Verified Solution • 2025–26 Session moderate

Let S =

\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm 1.

Then S represents :

A an ellipse whose eccentricity is ${1 \over {\sqrt {r + 1} }},$ where r > 1
B an ellipse whose eccentricity is ${2 \over {\sqrt {r + 1} }},$ where 0 < r < 1
C an ellipse whose eccentricity is ${2 \over {\sqrt {r - 1} }},$ where 0 < r < 1
D an ellipse whose eccentricity is $\sqrt {{2 \over {r + 1}}}$ , where r > 1

Accepted Answer

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

for r > 1,

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

e = \sqrt {1 - \left( {{{r - 1} \over {r + 2}}} \right)}

= \sqrt {{{\left( {r + 1} \right) - \left( {r - 1} \right)} \over {\left( {r + 1} \right)}}}

= \sqrt {{2 \over {r + 1}}} = \sqrt {{2 \over {r + 1}}}

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