JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

$A$ line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$ . Then the locus of the point, that divides the line segment $AB$ in the ratio $2 : 3$ , is a circle of radius :

Choose the correct answer:

Explanation

Part 1: Applying Cosine Rule

$\Delta OAP$ mein Cosine rule ka upyog karne par:

\cos A = \frac{OA^2 + AP^2 - OP^2}{2 \times OA \times AP}

Substitute values ( $\angle A = 60^\circ$ kyunki triangle equilateral hai):

\Rightarrow \cos 60^\circ = \frac{\lambda^2 + \left(\frac{2\lambda}{5}\right)^2 - OP^2}{2 \times \lambda \times \frac{2\lambda}{5}}

Part 2: Final Calculation

\Rightarrow \frac{1}{2} = \frac{\lambda^2 + \frac{4}{25}\lambda^2 - OP^2}{\frac{4}{5}\lambda^2}

\Rightarrow \frac{2}{5}\lambda^2 = \frac{29}{25}\lambda^2 - OP^2

\Rightarrow OP^2 = \frac{19}{25}\lambda^2 \quad \text{or} \quad OP = \frac{\sqrt{19}}{5}\lambda

Locus of point P:

OP = \frac{\sqrt{19}}{5}\lambda \text{ units}

Choose the correct answer:

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