The lines joining the origin to the point of intersection of The lines joining the origin to the point of intersection of `3x^2+m x y=4x+1=0` and `2x+y-1=0` are at right angles. Then which of the following is not a possible value of `m ?` `-4` (b) 4 (c) 7 (d) 3

Options

A. -4
B. 4
C. 7
D. 3

Explanation

To solve the problem, we need to find the values of \( m \) such that the lines joining the origin to the point of intersection of the given lines are at right angles. The given lines are:

1. \( 3x^2 + mxy - 4x = 0 \)
2. \( 2x + y - 1 = 0 \)

### Step 1: Find the point of intersection of the two lines

To find the intersection, we can express \( y \) from the second equation:

\[ y = 1 - 2x \]

Now, substitute this value of \( y \) into the first equation:

\[ 3x^2 + mx(1 - 2x) - 4x = 0 \]

### Step 2: Simplify the equation

Substituting \( y \) into the first equation gives:

\[ 3x^2 + mx - 2mx^2 - 4x = 0 \]

Combine like terms:

\[ (3 - 2m)x^2 + (m - 4)x = 0 \]

### Step 3: Factor the equation

Factoring out \( x \):

\[ x((3 - 2m)x + (m - 4)) = 0 \]

This gives us two cases:

1. \( x = 0 \) (which corresponds to the origin)
2. \( (3 - 2m)x + (m - 4) = 0 \)

### Step 4: Solve for \( x \)

From the second case:

\[ (3 - 2m)x = 4 - m \]

Thus,

\[ x = \frac{4 - m}{3 - 2m} \]

### Step 5: Find \( y \)

Substituting \( x \) back into \( y = 1 - 2x \):

\[ y = 1 - 2\left(\frac{4 - m}{3 - 2m}\right) = \frac{(3 - 2m) - 2(4 - m)}{3 - 2m} = \frac{3 - 2m - 8 + 2m}{3 - 2m} = \frac{-5}{3 - 2m} \]

### Step 6: Determine the slopes of the lines

The slopes of the lines joining the origin to the point \( \left(\frac{4 - m}{3 - 2m}, \frac{-5}{3 - 2m}\right) \) are:

\[ m_1 = \frac{-5}{4 - m} \quad \text{and} \quad m_2 = \frac{3 - 2m}{0} \]

For the lines to be perpendicular, the product of their slopes must equal -1:

\[ m_1 \cdot m_2 = -1 \]

### Step 7: Set up the equation for perpendicularity

Since \( m_2 \) is undefined (vertical line), we consider the condition for perpendicularity in terms of the slopes:

\[ \frac{-5}{4 - m} \cdot \frac{3 - 2m}{0} = -1 \]

### Step 8: Solve for \( m \)

We need to find the values of \( m \) that satisfy this condition. However, we are asked which of the given options is **not** a possible value of \( m \).

### Step 9: Evaluate the options

We can substitute each option into our derived equations to check if they yield valid results.

1. For \( m = -4 \):
\[ 3 - 2(-4) = 3 + 8 = 11 \quad \text{(valid)} \]

2. For \( m = 4 \):
\[ 3 - 2(4) = 3 - 8 = -5 \quad \text{(valid)} \]

3. For \( m = 7 \):
\[ 3 - 2(7) = 3 - 14 = -11 \quad \text{(valid)} \]

4. For \( m = 3 \):
\[ 3 - 2(3) = 3 - 6 = -3 \quad \text{(valid)} \]

### Conclusion

After evaluating all options, we find that all values yield valid results for the perpendicular condition except for one. Thus, the value that is **not** a possible value of \( m \) is:

**Answer: None of the provided values yield an invalid result.**