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The coefficient of correlations is r between X and Y having standard deviation σx and σy. The tangent of the between two lines of regression is:
1. \(\dfrac{1-r^2}{r}\)
2. \(\dfrac{1-r^2}{r} \sigma x \sigma y\)
3. \(\dfrac{1-r^2}{r} \dfrac{\sigma x \sigma y}{\sigma x + \sigma y}\)
4. \(\dfrac{1-r^2}{r} \dfrac{\sigma x \sigma y}{\sigma x^2 + \sigma y^2}\)

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Correct Answer - Option 4 : \(\dfrac{1-r^2}{r} \dfrac{\sigma x \sigma y}{\sigma x^2 + \sigma y^2}\)

Given

Coefficient of correlation r between X and Y = r(X, Y) = rxu = r

Standard deviation for X and Y = σx and σy

Calculation

The tangent of angle between two lines are given by

⇒ (1 - r)2( σx × σy)/r (σx2 + σ y2)

Equation of line of regression Y on X is

⇒ Y - y̅ = r × σy(X - x̅)/σx

The equation of line of regression X on Y is 

⇒ ⇒ X - x̅ = r × σx(Y - y̅ )/σy

Here slope of the line are

⇒ r × σyx and r × σxy

If θ is the angle between the two lines of regression

⇒ Tan θ =(r × σyx - r × σxx)/(1 + r × σyx × σxy)

⇒ (r2 - 1)(σx σy)/(r × σx2 × σ y2)

⇒ Tan θ = (1 - r2)(σx σy)/(r × σx2 × σ y2)

⇒ θ = tan-1{ (1 - r2)(σx σy)/(r × σx2 × σ y2)

Whenever two lines intersect, there are two angles between them, one is acute and another obtuse angle

If tan θ > 0 or o < θ < π/2 i.e θ is acute angle

.If  tan θ < 0 or π/2 < θ < 0 i.e θ is obtuse angle

∴ 0 < r2 < 1, the acute angle (θ1) and Obtuse angle (θ2) between two lines of regression are given by

θ1 = tan-1{ (1 - r2)(σx σy)/(r × σx2 × σ y2)

θ2 = tan-1{ (r2 - 1)(σx σy)/(r × σx2 × σ y2)

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