JEE PYQ: Hyperbola Question 7
Question 7 - 2020 (03 Sep Shift 1)
A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^2 + 4y^2 = 12$, then this hyperbola does not pass through which of the following points?
(1) $\left(\frac{1}{\sqrt{2}}, 0\right)$
(2) $\left(-\sqrt{\frac{3}{2}}, 1\right)$
(3) $\left(1, -\frac{1}{\sqrt{2}}\right)$
(4) $\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)$
Show Answer
Answer: (4)
Solution
Ellipse: $\frac{x^2}{4} + \frac{y^2}{3} = 1$, $c = \sqrt{4-3} = 1$. Foci: $(\pm 1, 0)$. Hyperbola: $2a = \sqrt{2}$, so $a = \frac{1}{\sqrt{2}}$. $c^2 = a^2 + b^2 \Rightarrow 1 = \frac{1}{2} + b^2 \Rightarrow b^2 = \frac{1}{2}$. Equation: $2x^2 - 2y^2 = 1$. Check option (4): $2 \cdot \frac{3}{2} - 2 \cdot \frac{1}{2} = 3 - 1 = 2 \neq 1$. So option (4) does not satisfy it.
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