The angle between the straight lines represented by the equation y² - xy - 6x² = 0 is

Concept:

Let two line represented by ax2 + 2hxy + by2 = 0 are

y = m1x

y = m2x

Where, 

\({m_1} + {m_2} = \frac{{ - 2h}}{b}\)      ----(i)

And \({m_1}{m_2} = \frac{a}{b}\)      ----(ii)

Let θ be angle between two lines

\(\tan θ = \frac{{ \pm( {m_1} - {m_2})}}{{1 + {m_1}{m_2}}}\)

∴ using (i) and (ii)

\(\tan θ = \frac{{ \pm 2\sqrt {{h^2} - ab} }}{{a + b}}\)  ----(iii)

Calculation:

Given equation is:

y2 - xy - 6x2 = 0 

Compare it with standard equation:

ax2 + 2hxy + by2 = 0

b = 1, a = - 6, h = -1/2

From equation (1);

\(\tan θ = \frac{{ \pm 2\sqrt {{(\frac{-1}{2})^2} - (-6)(1)} }}{{-6 + 1}}\)

\(\tan θ = \frac{{ \pm 2\sqrt {{(\frac{1}{4})} +6} }}{{-5}}\)

\(\tan θ = \frac{{ \pm 2\times\frac{5}{2} }}{{-5}}=\pm1\)

∴ θ = 45°