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If the acute angle between the lines ax^...

If the acute angle between the lines `ax^(2) + 2hxy + by^(2) = 0` is `60^(@)`, then show that `(a+3b)(3a+b) = 4h^(2)`.

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To solve the problem, we need to show that if the acute angle between the lines represented by the equation \( ax^2 + 2hxy + by^2 = 0 \) is \( 60^\circ \), then it follows that \( (a + 3b)(3a + b) = 4h^2 \). ### Step-by-Step Solution: 1. **Understanding the Equation**: The given equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines. The acute angle \( \theta \) between these lines can be expressed using the formula: \[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} ...
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