Question

Let \( F_1, F_2 \) \(\text{ be the foci of the hyperbola}\) \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, a > 0, \, b > 0, \] and let \( O \) be the origin. Let \( M \) be an arbitrary point on curve \( C \) and above the X-axis and \( H \) be a point on \( MF_1 \) such that \( MF_2 \perp F_1 F_2, \, M F_1 \perp OH, \, |OH| = \lambda |O F_2| \) with \( \lambda \in (2/5, 3/5) \), then the range of the eccentricity \( e \) is in:

• A. \( (1, \sqrt{7/3}) \)
• B. \( (\sqrt{7/3}, 2) \)
• C. \( (\sqrt{2}, \sqrt{3}) \)
• D. \( (\sqrt{3}, 2) \)

Hint:

The eccentricity of a hyperbola is determined by the relationship between the semi-major axis \( a \) and the semi-minor axis \( b \), and is always greater than 1.

Answer 1

The Correct Option is B

We are given the equation of a hyperbola and specific conditions involving the foci \( F_1, F_2 \), the point \( M \), and the relationship between the distances. We are tasked with finding the range of the eccentricity \( e \).

Step 1: Understanding the eccentricity of a hyperbola.
The eccentricity \( e \) of a hyperbola is given by: \[ e = \sqrt{1 + \frac{b^2}{a^2}} \]

Step 2: The relationship between distances.
We are given the conditions involving the distances from \( M \) and \( O \) to the foci. Using the relationship between the distances and the specific bounds for \( \lambda \), we can derive the bounds for \( e \).

Step 3: Conclusion.
After applying the conditions and solving the range for \( e \), we find that the range of the eccentricity is \( (\sqrt{7/3}, 2) \), corresponding to option (b). Thus, the correct answer is option (b).