A common tangent to the curves and does not pass through the fourth quadrant. If touches at and at ,

Q: JEE Main 2023 Conic Sections: A common tangent to the curves and does not pass through the fourth quadrant. If touches at and at ,...

The correct answer is 2. This solution will guide you through finding the common tangent to an ellipse and a hyperbola, determining the correct tangent based on a quadrant condition, and then calculating the required expression involving the points of tangency. --- **1. Key Concepts: Equations of Tangents with Given Slope** A fundamental concept in conic sections is the ability to write the equation of a tangent line if its slope is known. * For an ellipse , the equation of a tangent line with slope is given by: * For a

Question

A common tangent to the curves and does not pass through the fourth quadrant. If touches at and at , then is equal to .

Accepted Answer

This solution will guide you through finding the common tangent to an ellipse and a hyperbola, determining the correct tangent based on a quadrant condition, and then calculating the required expression involving the points of tangency.

1. Key Concepts: Equations of Tangents with Given Slope

A fundamental concept in conic sections is the ability to write the equation of a tangent line if its slope is known.

For an ellipse $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , the equation of a tangent line with slope $m$ is given by: $y = mx \pm \sqrt{a^2m^2 + b^2}$
For a hyperbola

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Solution

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