The equation of the line through the origin and perpendiculr to the line joining `(a,0)and (-a,0)` is

Options

A. `y=0`
B. `x=0`
C. `x=-a`
D. `y=-a`

Answer

To find the equation of the line through the origin and perpendicular to the line joining the points \( (a, 0) \) and \( (-a, 0) \), we can follow these steps:

### Step 1: Identify the points
The points given are \( A(a, 0) \) and \( B(-a, 0) \). These points lie on the x-axis.

**Hint:** Identify the coordinates of the points clearly.

### Step 2: Determine the slope of the line joining the points
The slope \( m \) of the line joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ m = \frac{0 - 0}{-a - a} = \frac{0}{-2a} = 0 \]

**Hint:** Remember that the slope of a horizontal line is 0.

### Step 3: Find the slope of the perpendicular line
If the slope of the original line \( m_1 = 0 \), then the slope of the line perpendicular to it \( m_2 \) can be found using the relationship:
\[ m_1 \cdot m_2 = -1 \]
Since \( m_1 = 0 \), the slope \( m_2 \) is undefined, which indicates a vertical line.

**Hint:** Recall that the slope of a vertical line is undefined.

### Step 4: Write the equation of the perpendicular line
A vertical line passing through the origin has the equation:
\[ x = 0 \]

**Hint:** A vertical line's equation is always in the form \( x = k \), where \( k \) is the x-coordinate of any point on the line.

### Final Answer
The equation of the line through the origin and perpendicular to the line joining \( (a, 0) \) and \( (-a, 0) \) is:
\[ \boxed{x = 0} \]