Question

Prove that the acute angle \( \theta \) between the lines represented by the equation \[ ax^2 + 2hxy + by^2 = 0 \] is \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|. \] Hence find the condition that the lines are coincident.

Hint:

For conic sections, coincident lines satisfy \( h^2 = ab \).

Answer 1

Step 1: Find Slopes of the Lines
The given equation represents two straight lines passing through the origin. The slopes are given by: \[ m_1, m_2 = \frac{- (h \pm \sqrt{h^2 - ab})}{b}. \] Step 2: Use the Angle Formula
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|. \] Substituting values: \[ \tan \theta = \left| \frac{ \frac{-h + \sqrt{h^2 - ab}}{b} - \frac{-h - \sqrt{h^2 - ab}}{b} }{1 + \left( \frac{-h + \sqrt{h^2 - ab}}{b} \right) \left( \frac{-h - \sqrt{h^2 - ab}}{b} \right) } \right|. \] \[ = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|. \] Step 3: Condition for Coincidence
For coincident lines, \( \theta = 0 \), which means: \[ h^2 - ab = 0 \quad \Rightarrow \quad h^2 = ab. \]