If the tangent at a point P on the parabola is parallel to the line and the tangents at the points Q

Q: JEE Main 2023 Conic Sections: If the tangent at a point P on the parabola is parallel to the line and the tangents at the points Q...

The correct answer is A. This problem integrates concepts from coordinate geometry, specifically properties of tangents to parabolas and ellipses, and the formula for the area of a triangle. We will systematically find the coordinates of points P, Q, and R, and then calculate the area of the triangle formed by them. --- ### **1. Finding Point P on the Parabola** **Key Concept:** For a parabola of the form , the equation of the tangent with slope is . The point of tangency corresponding to this tangent is . **Step-by-ste

Question

If the tangent at a point P on the parabola is parallel to the line and the tangents at the points Q and R on the ellipse are perpendicular to the line , then the area of the triangle PQR is :

Accepted Answer

This problem integrates concepts from coordinate geometry, specifically properties of tangents to parabolas and ellipses, and the formula for the area of a triangle. We will systematically find the coordinates of points P, Q, and R, and then calculate the area of the triangle formed by them.

1. Finding Point P on the Parabola

Key Concept: For a parabola of the form $y^2 = 4ax$ , the equation of the tangent with slope $m$ is $y = mx + \frac{a}{m}$ . The point of tangency corresponding to this tangent is $P\left(\frac{a}{m^2}, \frac{2a}{m}\right)$ .

Step-by-step Derivation:

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Solution

1. Finding Point P on the Parabola

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