Let PQ be a focal chord of the parabola y 2 = 4x such that it subtends an angle of at the point (3,

Q: JEE Main 2024 Conic Sections: Let PQ be a focal chord of the parabola y 2 = 4x such that it subtends an angle of at the point (3, ...

The correct answer is A. This problem is an excellent test of your understanding of conic sections, specifically parabolas and ellipses, and how to apply their properties in a coordinated manner. We will break down the problem into logical steps, focusing on defining and utilizing key concepts. --- **1. Understanding the Parabola and its Focal Chord** The given parabola is . **Key Concept:** The standard equation of a parabola is . Its focus is at and its directrix is . By comparing with , we identify . Therefore, the f

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Let PQ be a focal chord of the parabola y 2 = 4x such that it subtends an angle of at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse , . If e is the eccentricity of the ellipse E, then the value of is equal to :

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This problem is an excellent test of your understanding of conic sections, specifically parabolas and ellipses, and how to apply their properties in a coordinated manner. We will break down the problem into logical steps, focusing on defining and utilizing key concepts.

1. Understanding the Parabola and its Focal Chord

The given parabola is $y^2 = 4x$ .

Key Concept: The standard equation of a parabola is $y^2 = 4ax$ . Its focus is at $S(a,0)$ and its directrix is $x = -a$ . By comparing $y^2 = 4x$ with $y^2 = 4ax$ , we identify $a=1$ . Therefore, the focus of this parabola is $S(

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