If `theta` is the acute angle between the lines given by `ax^(2) + 2hxy + by^(2) = 0` then prove that `tan theta = |(2 sqrt (h^(2) - ab))/(a + b)|`. Hence find acute angle between the lines `2x^(2) + 7xy + 3y^(2) = 0`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understanding the Equation of the Pair of Lines
The equation of the pair of lines is given by:
\[ ax^2 + 2hxy + by^2 = 0 \]
where \( a \), \( b \), and \( h \) are constants. We need to find the acute angle \( \theta \) between these lines.

### Step 2: Finding the Slopes of the Lines
The slopes \( m_1 \) and \( m_2 \) of the lines can be derived from the quadratic equation:
\[ ax^2 + 2hxy + by^2 = 0 \]
Using the relationships:
- Sum of slopes: \( m_1 + m_2 = -\frac{2h}{b} \)
- Product of slopes: \( m_1 m_2 = \frac{a}{b} \)

### Step 3: Using the Formula for the Angle Between Two Lines
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

### Step 4: Substituting the Values
Substituting the values of \( m_1 + m_2 \) and \( m_1 m_2 \) into the formula:
1. Calculate \( m_1 - m_2 \):
\[ m_1 - m_2 = \sqrt{(m_1 + m_2)^2 - 4m_1 m_2} = \sqrt{\left(-\frac{2h}{b}\right)^2 - 4\left(\frac{a}{b}\right)} \]
\[ = \sqrt{\frac{4h^2}{b^2} - \frac{4a}{b}} = \frac{2\sqrt{h^2 - ab}}{b} \]

2. Calculate \( 1 + m_1 m_2 \):
\[ 1 + m_1 m_2 = 1 + \frac{a}{b} = \frac{b + a}{b} \]

### Step 5: Final Expression for \( \tan \theta \)
Now substituting these into the formula for \( \tan \theta \):
\[ \tan \theta = \frac{\frac{2\sqrt{h^2 - ab}}{b}}{\frac{b + a}{b}} = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Thus, we have proved that:
\[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]

### Step 6: Finding the Acute Angle for the Given Lines
Now, we need to find the acute angle between the lines given by:
\[ 2x^2 + 7xy + 3y^2 = 0 \]

Here, we identify:
- \( a = 2 \)
- \( h = \frac{7}{2} \)
- \( b = 3 \)

### Step 7: Substitute into the Formula
Now substitute these values into the formula:
1. Calculate \( h^2 - ab \):
\[ h^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
\[ ab = 2 \cdot 3 = 6 \]
\[ h^2 - ab = \frac{49}{4} - 6 = \frac{49}{4} - \frac{24}{4} = \frac{25}{4} \]

2. Substitute into the \( \tan \theta \) formula:
\[ \tan \theta = \left| \frac{2\sqrt{\frac{25}{4}}}{2 + 3} \right| = \left| \frac{2 \cdot \frac{5}{2}}{5} \right| = \left| \frac{5}{5} \right| = 1 \]

### Step 8: Finding the Angle
Since \( \tan \theta = 1 \), we have:
\[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \text{ radians} = 45^\circ \]

### Conclusion
The acute angle between the lines is \( 45^\circ \).

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