Find the angle between the straight lines joining the origin to the point of intersection of `3x^2+5x y-3y^2+2x+3y=0` and `3x-2y=1`

Explanation

To find the angle between the straight lines joining the origin to the point of intersection of the equations \(3x^2 + 5xy - 3y^2 + 2x + 3y = 0\) and \(3x - 2y = 1\), we can follow these steps:

### Step 1: Find the point of intersection
We need to solve the equations simultaneously. We can express \(y\) in terms of \(x\) from the linear equation \(3x - 2y = 1\).

\[ 2y = 3x - 1 \implies y = \frac{3x - 1}{2} \]

### Step 2: Substitute \(y\) in the first equation
Now, substitute \(y = \frac{3x - 1}{2}\) into the quadratic equation \(3x^2 + 5xy - 3y^2 + 2x + 3y = 0\).

\[ 3x^2 + 5x\left(\frac{3x - 1}{2}\right) - 3\left(\frac{3x - 1}{2}\right)^2 + 2x + 3\left(\frac{3x - 1}{2}\right) = 0 \]

### Step 3: Simplify the equation
Now, simplify the equation step by step:

1. Calculate \(5x\left(\frac{3x - 1}{2}\right) = \frac{15x^2 - 5x}{2}\)
2. Calculate \(-3\left(\frac{3x - 1}{2}\right)^2 = -3\left(\frac{9x^2 - 6x + 1}{4}\right) = -\frac{27x^2 - 18x + 3}{4}\)
3. Calculate \(3\left(\frac{3x - 1}{2}\right) = \frac{9x - 3}{2}\)

Now, substituting these back into the equation gives:

\[ 3x^2 + \frac{15x^2 - 5x}{2} - \frac{27x^2 - 18x + 3}{4} + 2x + \frac{9x - 3}{2} = 0 \]

### Step 4: Clear the fractions
Multiply through by 4 to eliminate the fractions:

\[ 12x^2 + 30x^2 - 10x - (27x^2 - 18x + 3) + 8x + 18x - 6 = 0 \]

Combine like terms:

\[ (12 + 30 - 27)x^2 + (-10 + 18 + 8)x + (3 - 6) = 0 \]

This simplifies to:

\[ 15x^2 + 16x - 3 = 0 \]

### Step 5: Solve for \(x\)
Now, we can use the quadratic formula to find \(x\):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 15 \cdot (-3)}}{2 \cdot 15} \]

Calculating the discriminant:

\[ 16^2 - 4 \cdot 15 \cdot (-3) = 256 + 180 = 436 \]

Thus,

\[ x = \frac{-16 \pm \sqrt{436}}{30} \]

### Step 6: Find corresponding \(y\)
Substituting \(x\) back into \(y = \frac{3x - 1}{2}\) will give us the corresponding \(y\) values.

### Step 7: Find the angle between the lines
The angle between the lines can be found using the formula:

\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Where \(m_1\) and \(m_2\) are the slopes of the lines. The slopes can be derived from the equations of the lines obtained from the quadratic equation.

### Step 8: Conclusion
Finally, calculate the angle \(\theta\) using the arctangent function.