Question

If the angle between the pair of lines given by the equation \(ax^2 + 4xy + 2y^2 = 0\) is \(45^\circ\), then the possible values of 'a' are

• A. \(-3\) or \(21\)
• B. \(-6 + 4\sqrt{3}\)
• C. \(-6 + 24\sqrt{2}\)
• D. do not exist

Hint:

When dealing with angles between lines in homogeneous equations, always use the angle formula to deduce relationships between coefficients.

Answer 1

The Correct Option is B

The general form of the equation for the pair of lines is \(ax^2 + 2hxy + by^2 = 0\). Here, \(h = 2\), \(b = 2\), and \(a\) varies. The angle \(\theta\) between the lines is given by: \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \] Substituting \(\theta = 45^\circ\) (so \(\tan 45^\circ = 1\)), and the values for \(h\) and \(b\): \[ 1 = \left| \frac{2\sqrt{4 - 2a}}{a + 2} \right| \] Solving this equation for \(a\): \[ 1 = \frac{2\sqrt{4 - 2a}}{a + 2} \] \[ (a + 2) = 2\sqrt{4 - 2a} \] \[ a^2 + 4a + 4 = 16 - 8a \] \[ a^2 + 12a - 12 = 0 \] Using the quadratic formula, \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(b = 12\), \(a = 1\), \(c = -12\): \[ a = \frac{-12 \pm \sqrt{144 + 48}}{2} \] \[ a = \frac{-12 \pm \sqrt{192}}{2} \] \[ a = -6 \pm 4\sqrt{3} \]