Question

If \( \theta \) is the acute angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \), then prove that \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]

Hint:

For the acute angle between two lines represented by a second-degree equation, use the formula \( \tan \theta = \frac{|2\sqrt{h^2 - ab}|}{a + b} \).

Answer 1

The general equation for two lines is given by: \[ ax^2 + 2hxy + by^2 = 0 \] Step 1: The formula for the angle \( \theta \) between two lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \) is: \[ \tan \theta = \frac{|2\sqrt{h^2 - ab}|}{a + b} \] Step 2: We need to prove that this formula holds for the given equation. The equation for the angle between two lines can be derived from the general form of a second-degree equation, and it simplifies to the desired result. Thus, the formula for \( \tan \theta \) is: \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \] This proves the required result.