Analyzing Straight Line Equations

We are given the equations of two straight lines:

1. Line 1: x + 2y + 4 = 0
2. Line 2: -4x + 2y - 3 = 0

We need to determine the relationship between these lines based on the given options.

Checking if Straight Lines are Parallel

Two straight lines are parallel if their slopes are equal. The slope of a line with the equation $Ax + By + C = 0$ is given by $m = -\frac{A}{B}$, provided $B \neq 0$.

• For Line 1 ($x + 2y + 4 = 0$), $A_1 = 1$ and $B_1 = 2$. The slope is $m_1 = -\frac{1}{2}$.
• For Line 2 ($-4x + 2y - 3 = 0$), $A_2 = -4$ and $B_2 = 2$. The slope is $m_2 = -\frac{-4}{2} = \frac{4}{2} = 2$.

Since $m_1 = -\frac{1}{2}$ and $m_2 = 2$, the slopes are not equal ($m_1 \neq m_2$). Therefore, the lines are not parallel.

This means option 1 ("they are parallel") is incorrect.

Checking if Straight Lines Pass Through the Origin

A straight line passes through the origin $(0,0)$ if substituting $x=0$ and $y=0$ into the equation satisfies the equation.

• For Line 1 ($x + 2y + 4 = 0$), substitute $x=0, y=0$:
  (0) + 2(0) + 4 = 0 + 0 + 4 = 4
  Since $4 \neq 0$, Line 1 does not pass through the origin.
• For Line 2 ($-4x + 2y - 3 = 0$), substitute $x=0, y=0$:
  -4(0) + 2(0) - 3 = 0 + 0 - 3 = -3
  Since $-3 \neq 0$, Line 2 does not pass through the origin.

Neither line passes through the origin.

This means option 2 ("both are passing through the origin") is incorrect, and option 3 ("one is passing through the origin") is also incorrect.

Evaluating Options

Based on our analysis:

• Option 1: "they are parallel" - Incorrect (slopes are $-\frac{1}{2}$ and $2$).
• Option 2: "both are passing through the origin" - Incorrect (neither line passes through $(0,0)$).
• Option 3: "one is passing through the origin" - Incorrect (neither line passes through $(0,0)$).
• Option 4: "More than one of the above" - Incorrect, as options 1, 2, and 3 are all incorrect.
• Option 5: "None of the above" - This option is correct because none of the conditions described in options 1, 2, or 3 are true for the given pair of lines.

The correct answer is that none of the given statements are true for the two straight lines.