Let a conic pass through the point and , be any point on . Let the slope of the line touching the co

Q: JEE Main 2024 Conic Sections: Let a conic pass through the point and , be any point on . Let the slope of the line touching the co...

The correct answer is 5. This problem requires us to identify a conic section from a given geometric property, determine its specific equation using a point it passes through, and finally calculate a focal distance. The journey involves setting up and solving a differential equation, recognizing the standard form of a conic, and applying its properties. --- **1. Problem Analysis and Initial Setup** We are given the following information about a conic : * It passes through the point . * For any point on (with ), there's

Question

Let a conic pass through the point and , be any point on . Let the slope of the line touching the conic only at a single point be half the slope of the line joining the points and . If the focal distance of the point on is , then equals .

Accepted Answer

This problem requires us to identify a conic section from a given geometric property, determine its specific equation using a point it passes through, and finally calculate a focal distance. The journey involves setting up and solving a differential equation, recognizing the standard form of a conic, and applying its properties.

1. Problem Analysis and Initial Setup

We are given the following information about a conic $C$ :

It passes through the point $(4,-2)$ .
For any point $P(x, y)$ on $C$ (with $x \geq 3$ ), there's a specific relationship between the slope of the tangent line at $P$ and the slope of the line joining $P$ to a fixed point $Q(3,-5)$ . *

Question

Solution

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