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) = r_{xu} = r

Standard deviation for X and Y = σ_{x} and σ_{y}

**Calculation**

The tangent of angle between two lines are given by

⇒ (1 - r)^{2}( σx × σ_{y})/r (σ_{x}^{2} + σ _{y}^{2})

Equation of line of regression Y on X is

⇒ Y - y̅ = r × σ_{y}(X - x̅)/σ_{x}

The equation of line of regression X on Y is

⇒ ⇒ X - x̅ = r × σ_{x}(Y - y̅ )/σ_{y}

Here slope of the line are

⇒ r × σ_{y}/σ_{x} and r × σ_{x}/σ_{y}

If θ is the angle between the two lines of regression

⇒ Tan θ =(r × σy/σ_{x} - r × σx/σ_{x})/(1 + r × σy/σ_{x} × σx/σ_{y})

⇒ (r^{2} - 1)(σ_{x} σ_{y})/(r × σ_{x}^{2} × σ _{y}^{2})

⇒ Tan θ = (1 - r^{2})(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 < r**^{2} < 1, the acute angle (θ_{1}) and Obtuse angle (θ_{2}) between two lines of regression are given by

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

**θ**_{2} = tan-1{ (r^{2} - 1)(σx σy)/(r × σx2 × σ y2)