Let A (sec , 2tan ) and B (sec , 2tan ), where + = /2, be two points on the hyperbola 2x 2 y 2 = 2.

Q: JEE Main 2023 Conic Sections: Let A (sec , 2tan ) and B (sec , 2tan ), where + = /2, be two points on the hyperbola 2x 2 y 2 = 2. ...

The correct answer is 2. This problem tests a fundamental understanding of curve definitions, parametric forms, and the consistency of given conditions in a mathematical problem. Often, in competitive exams, a seemingly complex setup might hide a critical inconsistency or simplification that, once identified, makes the problem much easier. Let's break down the problem step-by-step. --- **1. Verifying the Parametric Points on the Hyperbola** The first and most crucial step is to check if the given parametric points actua

Question

Let A (sec , 2tan ) and B (sec , 2tan ), where + = /2, be two points on the hyperbola 2x 2 y 2 = 2. If ( , ) is the point of the intersection of the normals to the hyperbola at A and B, then (2 ) 2 is equal to .

Accepted Answer

This problem tests a fundamental understanding of curve definitions, parametric forms, and the consistency of given conditions in a mathematical problem. Often, in competitive exams, a seemingly complex setup might hide a critical inconsistency or simplification that, once identified, makes the problem much easier.

Let's break down the problem step-by-step.

1. Verifying the Parametric Points on the Hyperbola

The first and most crucial step is to check if the given parametric points actually lie on the specified hyperbola. If they don't, or if they only do under specific, restrictive conditions, this insight will guide our entire approach.

The given hyperbola is $2x^2 - y^2 = 2$ . Dividing by 2, we get the standard form:

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Solution

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