The straight lines joining origin to the points of intersection of the straight line `3x-y-2=0` and the curve `7x^(2)-4xy+8y^(2)+2x-4y-8=0` are inclined to each other at angle

Options

A. `pi//4`
B. `pi//3`
C. `pi//2`
D. none of these

Answer

To solve the problem, we need to find the angle between the straight lines joining the origin to the points of intersection of the given straight line and the curve. Here’s a step-by-step solution:

### Step 1: Find the points of intersection
We start with the straight line equation:
\[ 3x - y - 2 = 0 \]
From this, we can express \( y \) in terms of \( x \):
\[ y = 3x - 2 \]

Next, we substitute this expression for \( y \) into the curve equation:
\[ 7x^2 - 4xy + 8y^2 + 2x - 4y - 8 = 0 \]
Substituting \( y = 3x - 2 \):
\[ 7x^2 - 4x(3x - 2) + 8(3x - 2)^2 + 2x - 4(3x - 2) - 8 = 0 \]

### Step 2: Simplify the equation
Now we simplify the equation:
1. Calculate \( 8(3x - 2)^2 \):
\[ 8(9x^2 - 12x + 4) = 72x^2 - 96x + 32 \]

2. Substitute back into the equation:
\[ 7x^2 - 12x^2 + 72x^2 + 2x - 12x + 8 - 8 = 0 \]
Combine like terms:
\[ (7 - 12 + 72)x^2 + (2 - 12)x + 0 = 0 \]
\[ 67x^2 - 10x = 0 \]

### Step 3: Factor the equation
Factoring out \( x \):
\[ x(67x - 10) = 0 \]
This gives us:
1. \( x = 0 \)
2. \( 67x - 10 = 0 \Rightarrow x = \frac{10}{67} \)

### Step 4: Find corresponding y values
For \( x = 0 \):
\[ y = 3(0) - 2 = -2 \]
So one point is \( (0, -2) \).

For \( x = \frac{10}{67} \):
\[ y = 3\left(\frac{10}{67}\right) - 2 = \frac{30}{67} - \frac{134}{67} = -\frac{104}{67} \]
So the second point is \( \left(\frac{10}{67}, -\frac{104}{67}\right) \).

### Step 5: Find slopes of the lines
The slopes of the lines joining the origin to these points are:
1. For \( (0, -2) \):
\[ m_1 = \frac{-2 - 0}{0 - 0} \text{ (undefined, vertical line)} \]

2. For \( \left(\frac{10}{67}, -\frac{104}{67}\right) \):
\[ m_2 = \frac{-\frac{104}{67} - 0}{\frac{10}{67} - 0} = -\frac{104}{10} = -\frac{52}{5} \]

### Step 6: Find the angle between the lines
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Since \( m_1 \) is undefined (vertical line), the angle between a vertical line and any other line is \( 90^\circ \).

### Conclusion
Thus, the angle between the two lines joining the origin to the points of intersection is:
\[ \theta = 90^\circ \]