Let be the foci of the hyperbola ( ), and let be the origin. Let be an arbitrary point on curve above the -axis and be a point on such that , , and with . Then the range of the eccentricity is:

Explanation: Solution Coordinates: , , Since , Slope of : Equation of line : Length of (Distance from to ): Relation with : Range Calculation: For : For : Final Answer: The range of is . The correct option is (A) .

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NIMCET 2025 — Mathematics PYQ

NIMCET | Mathematics | 2025

Let $F_{1}, F_{2}$ 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 $C$ above the $X$ -axis and $H$ be a point on $MF_{1}$ such that $MF_{2} \perp F_{1}F_{2}$ , $MF_{1} \perp OH$ , and $|OH| = \lambda |OF_{2}|$ with $\lambda \in (2/5, 3/5)$ . Then the range of the eccentricity $e$ is:

Choose the correct answer: