Find the value of h, if the measure of the angle between the lines `3x^(2) + 2hxy + 2y^(2) = 0` is `45^(@)`.

Answer

To find the value of \( h \) such that the angle between the lines given by the equation \( 3x^2 + 2hxy + 2y^2 = 0 \) is \( 45^\circ \), we can follow these steps:

### Step 1: Identify coefficients
The general form of the equation of a pair of straight lines is given by:
\[ ax^2 + 2hxy + by^2 = 0 \]
From the given equation \( 3x^2 + 2hxy + 2y^2 = 0 \), we can identify:
- \( a = 3 \)
- \( b = 2 \)

### Step 2: Use the angle formula
The formula for the tangent of the angle \( \theta \) between the two lines is:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
Given that \( \theta = 45^\circ \), we know that:
\[ \tan 45^\circ = 1 \]
Thus, we can set up the equation:
\[ 1 = \frac{2\sqrt{h^2 - (3)(2)}}{3 + 2} \]

### Step 3: Simplify the equation
Substituting the values of \( a \) and \( b \):
\[ 1 = \frac{2\sqrt{h^2 - 6}}{5} \]
Now, multiply both sides by 5:
\[ 5 = 2\sqrt{h^2 - 6} \]

### Step 4: Isolate the square root
Next, divide both sides by 2:
\[ \frac{5}{2} = \sqrt{h^2 - 6} \]

### Step 5: Square both sides
Now, square both sides to eliminate the square root:
\[ \left(\frac{5}{2}\right)^2 = h^2 - 6 \]
This simplifies to:
\[ \frac{25}{4} = h^2 - 6 \]

### Step 6: Solve for \( h^2 \)
Add 6 to both sides:
\[ h^2 = \frac{25}{4} + 6 \]
Convert 6 into a fraction with a denominator of 4:
\[ h^2 = \frac{25}{4} + \frac{24}{4} = \frac{49}{4} \]

### Step 7: Take the square root
Now, take the square root of both sides:
\[ h = \pm \frac{7}{2} \]

### Final Answer
Thus, the values of \( h \) are:
\[ h = \frac{7}{2} \quad \text{or} \quad h = -\frac{7}{2} \]

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