Find the equations of the straight lines joining the origin to the points of intersection of the line $x - y = 2$ and the curve $5 x^{2} + 12 x y - 8 y^{2} + 8 x - 4 y + 12 = 0$ and find the angle between them. Also show that these lines are equally inclined to the axes.

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Solution

From the equation of the line we have $x - y = 2$
or $\frac{x - y}{2} = 1$ .. $(1)$
Equation of the curve
$5 x^{2} + 12 x y - 8 y^{2} + 4 (2 x - y) + 12 = 0$
$5 x^{2} + 12 x y - 8 y^{2} + 4 (2 x - y) \cdot 1 + 12 \cdot 1^{2} = 0$ [Make it homogeneous]
Equation of the required lines is
$5 x^{2} + 12 x y - 8 y^{2} + 4 (2 x - y) (\frac{x - y}{2}) + 12 {(\frac{x - y}{2})}^{2} = 0$ [by $(1)$ ]
or $[5 x^{2} + 12 x y - 8 y^{2}) + 2 (2 x^{2} - 3 x y + y^{2}) + 3 (x^{2} - 2 x y + y^{2}) = 0$
or $12 x^{2} + 0 x y - 3 y^{2} = 0$ or $y = \pm 2 x$ .
If $tan θ = 2,$ then $- 2 = - tan θ = tan (π - θ)$ .
They are clearly equally inclined to axes.
Angle between them is $π - 2 θ = π - 2 {tan}^{- 1} 2$ .

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Find the equations of the straight lines joining the origin to the points of intersection of the line x−y=2 and the curve 5x2+12xy−8y2+8x−4y+12=0 and find the angle between them. Also show that these lines are equally inclined to the axes.

Find the equations of the straight lines joining the origin to the points of intersection of the line $x - y = 2$ and the curve $5 x^{2} + 12 x y - 8 y^{2} + 8 x - 4 y + 12 = 0$ and find the angle between them. Also show that these lines are equally inclined to the axes.