Let P(4, –4) and Q(9, 6) be two points on the parabola, y 2 = 4x and let x be any point on the arc P

Q: JEE Main 2021 Conic Sections: Let P(4, –4) and Q(9, 6) be two points on the parabola, y 2 = 4x and let x be any point on the arc P...

The correct answer is A. This problem requires us to find the maximum area of a triangle , where the base PQ is fixed, and the third vertex X moves along a specific arc of a parabola. The core concept for such optimization problems is rooted in geometry. --- **1. Key Concept: Maximizing the Area of a Triangle with a Fixed Base** For a triangle with a **fixed base** (in this case, the line segment PQ), its area is maximized when the **height** from the third vertex (X) to that base is maximum. Geometrically, this conditi

Question

Let P(4, –4) and Q(9, 6) be two points on the parabola, y 2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of PXQ is maximum. Then this maximum area (in sq. units) is :

Accepted Answer

This problem requires us to find the maximum area of a triangle $\Delta PXQ$ , where the base PQ is fixed, and the third vertex X moves along a specific arc of a parabola. The core concept for such optimization problems is rooted in geometry.

1. Key Concept: Maximizing the Area of a Triangle with a Fixed Base

For a triangle with a fixed base (in this case, the line segment PQ), its area is maximized when the height from the third vertex (X) to that base is maximum.

Geometrically, this condition is met when the third vertex X is farthest from the line segment PQ. This occurs precisely when the tangent to the curve at point X is parallel to the base PQ.

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