A line is a common tangent to the circle (x 3) 2 + y 2 = 9 and the parabola y 2 = 4x. If the two poi

Q: JEE Main 2021 Conic Sections: A line is a common tangent to the circle (x 3) 2 + y 2 = 9 and the parabola y 2 = 4x. If the two poi...

The correct answer is 3. This problem involves finding a common tangent line to a parabola and a circle, identifying their respective points of contact in the first quadrant, and then performing a specific calculation based on these coordinates. We will leverage the standard forms of tangent equations and the condition for tangency. --- **1. Key Concepts and Formulas** * **Equation of a Tangent to a Parabola:** For a parabola of the standard form , the equation of a tangent line with slope is given by: The coordinates o

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A line is a common tangent to the circle (x 3) 2 + y 2 = 9 and the parabola y 2 = 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to .

Accepted Answer

This problem involves finding a common tangent line to a parabola and a circle, identifying their respective points of contact in the first quadrant, and then performing a specific calculation based on these coordinates. We will leverage the standard forms of tangent equations and the condition for tangency.

1. Key Concepts and Formulas

Equation of a Tangent to a Parabola: For a parabola of the standard form $y^2 = 4Ax$ , the equation of a tangent line with slope $m$ is given by: $y = mx + \frac{A}{m}$ The coordinates of the point of contact $(x_P, y_P)$ for this tangent on the parabola are:

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