Find the equation to the straight line passing through
the origin and perpendicular to `x+2y=4`

Answer

To find the equation of the straight line passing through the origin and perpendicular to the line given by the equation \( x + 2y = 4 \), we can follow these steps:

### Step 1: Find the slope of the given line
To find the slope of the line \( x + 2y = 4 \), we first rewrite it in the slope-intercept form \( y = mx + c \).

Starting with:
\[ x + 2y = 4 \]

We can isolate \( y \):
\[ 2y = -x + 4 \]
\[ y = -\frac{1}{2}x + 2 \]

From this equation, we can see that the slope \( m_1 \) of the given line is \( -\frac{1}{2} \).

### Step 2: Determine the slope of the perpendicular line
If two lines are perpendicular, the product of their slopes is \( -1 \). Let \( m_2 \) be the slope of the line we want to find. Thus, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting \( m_1 = -\frac{1}{2} \):
\[ -\frac{1}{2} \cdot m_2 = -1 \]

To find \( m_2 \), we can rearrange this equation:
\[ m_2 = \frac{-1}{-\frac{1}{2}} = 2 \]

### Step 3: Write the equation of the line through the origin
Since the line passes through the origin (0, 0), the equation of the line can be expressed as:
\[ y = mx \]
Substituting \( m = 2 \):
\[ y = 2x \]

### Final Answer
Thus, the equation of the straight line passing through the origin and perpendicular to the line \( x + 2y = 4 \) is:
\[ \boxed{y = 2x} \]