JEE PYQ: Hyperbola Question 19
Question 19 - 2019 (10 Jan Shift 2)
Let $S = \left{(x,y) \in \mathbb{R}^2 : \frac{y^2}{1+r} - \frac{x^2}{1-r} = 1\right}$, where $r \neq \pm 1$. Then S represents:
(1) a hyperbola whose eccentricity is $\frac{2}{\sqrt{1-r}}$, when $0 < r < 1$
(2) an ellipse whose eccentricity is $\sqrt{\frac{2}{r+1}}$, when $r > 1$
(3) a hyperbola whose eccentricity is $\frac{2}{\sqrt{r+1}}$, when $0 < r < 1$
(4) an ellipse whose eccentricity is $\frac{1}{\sqrt{r+1}}$, when $r > 1$
Show Answer
Answer: (2)
Solution
When $r > 1$: $1-r < 0$, so $\frac{y^2}{1+r} + \frac{x^2}{r-1} = 1$ (ellipse with $a^2 = 1+r$, $b^2 = r-1$ since $1+r > r-1$). $e^2 = 1 - \frac{r-1}{r+1} = \frac{2}{r+1}$, so $e = \sqrt{\frac{2}{r+1}}$. When $0 < r < 1$: both denominators positive, so it’s a hyperbola: $\frac{y^2}{1+r} - \frac{x^2}{1-r} = 1$, $e^2 = 1 + \frac{1-r}{1+r} = \frac{2}{1+r}$, $e = \sqrt{\frac{2}{r+1}}$.
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