If `theta` is the measure of the acute angle between the lines represented by equation `ax^(2)+2hx+by^(2)=0`, then prove that `tan theta =|(2sqrt(h^(2)-ab))/(a+b)|` where `a+b ne 0 and ne 0.` Find the condition for coincident lines.

Explanation

To prove that \( \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \) for the acute angle \( \theta \) between the lines represented by the equation \( ax^2 + 2hx + by^2 = 0 \), we can follow these steps:

### Step 1: Identify the form of the equation
The given equation is \( ax^2 + 2hx + by^2 = 0 \). This represents a pair of straight lines if it can be factored.

### Step 2: Use the condition for the pair of lines
For the equation \( ax^2 + 2hx + by^2 = 0 \) to represent two lines, the discriminant must be non-negative. The discriminant \( D \) for the quadratic in \( x \) is given by:
\[
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