Let S = x , y ∈ R 2 : y 2 1 + r - x 2 1 - r = 1 , where r ≠ ± 1 . Then S represents:

Given y 2 1 + r - x 2 1 - r = 1 If r ∈ 0 , 1 then this curve is a hyperbola whose eccentricity is 1 + 1 - r 1 + r = 2 r + 1 . If r > 1 , then this curve is an ellipse whose eccentricity is 1 - r - 1 r + 1 = 2 r + 1 .

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Mathematics - Ellipse Question with Solution | TestHub

MathematicsEllipseGeneralMedium2 minPYQ_2019

MathematicsMediumstatement

Let $S = \{(x, y) \in R^{2} : \frac{y^{2}}{1 + r} - \frac{x^{2}}{1 - r} = 1\},$ where $r \neq \pm 1 .$ Then $S$ represents:

Options:

Answer:

Solution:

Given $\frac{y^{2}}{1 + r} - \frac{x^{2}}{1 - r} = 1$

If $r \in (0, 1)$ then this curve is a hyperbola whose eccentricity is $\sqrt{1 + \frac{1 - r}{1 + r}} = \sqrt{\frac{2}{r + 1}}$ .

If $r > 1$ , then this curve is an ellipse whose
eccentricity is $\sqrt{1 - \frac{r - 1}{r + 1}} = \sqrt{\frac{2}{r + 1}}$ .

Stream:JEESubject:MathematicsTopic:EllipseSubtopic:General

⏱ 2mℹ️ Source: PYQ_2019

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