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Question: Answered & Verified by Expert
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
MathematicsCircleJEE MainJEE Main 2023 (10 Apr Shift 1)
Options:
  • A 35λ
  • B 23λ
  • C 195λ
  • D 197λ
Solution:
1519 Upvotes Verified Answer
The correct answer is: 195λ

Given,

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,

Now taking the points on the circle of radius λ as Bλcosθ1,λsinθ1 & Aλcosθ2λsinθ2 and taking the point Ph,k which divides the line segment in AB of length λ in 2:3

Now plotting the diagram we get,

Now, let O be the origin and radius of circle is λ and  AB=λ and using distance formula we get,

AB=λ=λcosθ1-λcosθ22+λsinθ1-λsinθ22

1=2-2 cosθ1-θ2

cosθ1-θ2=12

Now using section formula we get,

h=2λ cosθ1+3λ cosθ25 and  k=2λ sinθ1+3λ sinθ25

Now squaring and adding above two value we get,

h2+k2=λ2254+9+12cosθ1-θ2

h2+k2=λ225·19

Hence, Radius =λ519

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