A line segment AB of length moves such that the points A and B remain on the periphery of a circle of radius . Then the locus of the point that divides the line segment AB in the ratio 2 : 3, is a circle of radius [2023]

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Solution

Q.
A line segment AB of length $λ$ moves such that the points A and B remain on the periphery of a circle of radius $λ$ . Then the locus of the point that divides the line segment AB in the ratio 2 : 3, is a circle of radius [2023]

1 $\frac{\sqrt{19}}{5} λ$

2 $\frac{2}{3} λ$

3 $\frac{\sqrt{19}}{7} λ$

4 $\frac{3}{5} λ$

Ans.
(1)

$∠ O A P = 60 °$ $[∵ O A = O B = A B = λ]$

$A P = \frac{2 λ}{5} (Given)$

$\cos 60 ° = \frac{O A^{2} + A P^{2} - O P^{2}}{2 \cdot O A \cdot A P}$

$\Rightarrow$ $\frac{1}{2} = \frac{λ^{2} + \frac{4 λ^{2}}{25} - O P^{2}}{2 λ \cdot \frac{2 λ}{5}}$

$\Rightarrow \frac{2 λ^{2}}{5} = \frac{29 λ^{2}}{25} - O P^{2} \Rightarrow O P^{2} = \frac{29}{25} λ^{2} - \frac{2}{5} λ^{2} = \frac{19}{25} λ^{2}$

Required locus is $x^{2} + y^{2} = \frac{19}{25} λ^{2}$

$∴$ $Radius = \frac{\sqrt{19}}{5} λ$

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