Find the equations of the straight lines bisecting the angles between the pairs of straight lines : $12 x^{2} - 7 x y - 12 y^{2} = 0; 71 x^{2} + 94 x y - 71 y^{2} = 0; x^{2} - y^{2} = 0$

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Solution

$For a pair of straight lines given by the equation a x^{2} + 2 h x y + b y^{2} = 0, equation of the angle bisectors is given by \frac{x^{2} - y^{2}}{a - b} = \frac{x y}{h}$

$Equation of the pair of straight lines is 12 x^{2} - 7 x y - 12 y^{2} = 0 a = 12, b = - 12 and h = \frac{- 7}{2} ∴ Equation of the angle bisector is \frac{x^{2} - y^{2}}{24} = \frac{- 2 x y}{7} or 7 x^{2} + 48 x y - 7 y^{2} = 0$

$Equation of the pair of straight lines is 71 x^{2} + 94 x y - 71 y^{2} = 0 a = 71, b = - 71 and h = 47 ∴ Equation of the angle bisector is \frac{x^{2} - y^{2}}{142} = \frac{x y}{47} or 47 x^{2} - 142 x y - 47 y^{2} = 0$

$Equation of the pair of straight lines is x^{2} - y^{2} = 0 a = 1, b = - 1 and h = 0 ∴ Equation of the angle bisector is \frac{x^{2} - y^{2}}{2} = \frac{x y}{0} or x y = 0$

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