Suppose rxy is the correlation coefficient between two variables X and Y
where s.d.(X) = s.d.(Y). If θ is the angle between the two regression lines of Y on X and X on Y then:

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SSC CGL Tier-II ( JSO ) 2016 Official Paper ( Held On : 2 Dec 2016 )
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Answer (Detailed Solution Below)

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PYST 1:SSC CGL 2023 - Quantitative Aptitude(Held On:14 Jul 2023 Shift 1)
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Detailed Solution

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Formula

If θ be the angle between the regression lines then,

Tan θ = [(1 - r2)/r] × (σx ×  σy)/(σx2 + σy2)

Explanation

According to question

⇒ σx = σy

⇒ Tanθ = [(1 - r2/r] × (σx2/(2σx)

⇒ Tanθ = [(1 - r2/r] × 1/2

⇒ Tan θ = (1 - r2/2r

⇒ Tanθ = Sinθ/cosθ

⇒ Tanθ = Sinθ × cosθ 

⇒ Sinθ = Tanθ /secθ 

⇒ Sinθ = Tanθ/√(1 + tan2θ)

⇒ Sinθ = [(1 - r2/r]/√(1 + (1 - r2/2r)2)

⇒ Sinθ = (1 - r2)/√[(1 - r2)2 + 4r2]

∴ 

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