Question

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?

• A. \(\left(1, -\frac{1}{\sqrt{2}}\right)\)
• B. \(\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)\)
• C. \(\left(-\sqrt{\frac{3}{2}}, 1\right)\)
• D. \(\left(\frac{1}{\sqrt{2}}, 0\right)\)

Hint:

Find the foci of the ellipse, use 2a = \(\sqrt{2}\) and \(b^2 = c^2 - a^2\) to get the hyperbola, then test each point.

Answer 1

The Correct Option is B

The ellipse is \(\frac{x^2}{4} + \frac{y^2}{3} = 1\), so \(a^2 = 4, b^2 = 3\) and \(e = \sqrt{1 - \frac{3}{4}} = \frac{1}{2}\). Foci: \((\pm ae, 0) = (\pm 1, 0)\).
For the hyperbola, \(c = 1\) and the transverse axis \(2a' = \sqrt{2}\), so \(a'^2 = \frac{1}{2}\).
\(b'^2 = c^2 - a'^2 = 1 - \frac{1}{2} = \frac{1}{2}\).
Hyperbola: \(\frac{x^2}{1/2} - \frac{y^2}{1/2} = 1\), i.e. \(x^2 - y^2 = \frac{1}{2}\).
Test the points:
(1) \(1 - \frac{1}{2} = \frac{1}{2}\) satisfied.
(2) \(\frac{3}{2} - \frac{1}{2} = 1 \ne \frac{1}{2}\) not satisfied.
(3) \(\frac{3}{2} - 1 = \frac{1}{2}\) satisfied.
(4) \(\frac{1}{2} - 0 = \frac{1}{2}\) satisfied.
So the hyperbola does not pass through \(\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)\).
Answer: (2) \(\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)\)