Let be a point on the parabola and PQ be a focal chord of the parabola. If M and N are the foot of p

Q: JEE Main 2023 Conic Sections: Let be a point on the parabola and PQ be a focal chord of the parabola. If M and N are the foot of p...

The correct answer is A. This problem requires a strong understanding of parabola properties, including its definition, focal chords, parametric representation, and basic coordinate geometry for calculating areas. We will systematically break down the problem to find the required area. --- **1. Determine the Parabola's Parameters: 'a', Focus, and Directrix** **Key Concept:** For a standard parabola , a point lying on it must satisfy its equation. The focus is at and the directrix is the line . **Step-by-step working:**

Question

Let be a point on the parabola and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

Accepted Answer

This problem requires a strong understanding of parabola properties, including its definition, focal chords, parametric representation, and basic coordinate geometry for calculating areas. We will systematically break down the problem to find the required area.

1. Determine the Parabola's Parameters: 'a', Focus, and Directrix

Key Concept: For a standard parabola $y^2 = 4ax$ , a point $(x_1, y_1)$ lying on it must satisfy its equation. The focus is at $S(a,0)$ and the directrix is the line $x=-a$ .

Step-by-step working: The given point is $P(4, 4\sqrt{3})$ , and it lies on the parabola $y^2 =

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