If two tangents drawn from a point ( , ) lying on the ellipse 25x 2 + 4y 2 = 1 to the parabola y 2 =

Q: JEE Main 2024 Conic Sections: If two tangents drawn from a point ( , ) lying on the ellipse 25x 2 + 4y 2 = 1 to the parabola y 2 =...

The correct answer is 25. This solution will guide you through the process of solving the given problem, emphasizing the underlying concepts and detailed algebraic steps. We will derive the relationships between , , and the slopes of the tangents, and then substitute these relationships into the final expression. --- **1. Key Concepts and Formulas** * **Equation of a Tangent to a Parabola:** For a parabola of the form , the equation of a tangent with slope is given by . * *Tip:* This is a standard form. Make sure you can

Question

If two tangents drawn from a point ( , ) lying on the ellipse 25x 2 + 4y 2 = 1 to the parabola y 2 = 4x are such that the slope of one tangent is four times the other, then the value of (10 + 5) 2 + (16 2 + 50) 2 equals .

Accepted Answer

This solution will guide you through the process of solving the given problem, emphasizing the underlying concepts and detailed algebraic steps. We will derive the relationships between $\alpha$ , $\beta$ , and the slopes of the tangents, and then substitute these relationships into the final expression.

1. Key Concepts and Formulas

Equation of a Tangent to a Parabola: For a parabola of the form $y^2 = 4ax$ $y^{2} = 4 a x$ , the equation of a tangent with slope $m$ $m$ is given by $y = mx + \frac{a}{m}$ $y = m x + \frac{a}{m}$ .
- Tip: This is a standard form. Make sure you can derive it by finding the derivative $\frac{dy}{dx}$ and setting it equal to $m$ .
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