A line through origin meets the line `x = 3y + 2` at right angles at point X. Find the co-ordinates of X. 

Answer

To solve the problem step by step, we will find the coordinates of the point X where the line through the origin meets the line \( x = 3y + 2 \) at right angles.

### Step 1: Identify the equations of the lines
The line through the origin can be represented as:
\[ y = mx \]
where \( m \) is the slope of the line.

The given line is:
\[ x = 3y + 2 \]

### Step 2: Convert the given line to slope-intercept form
To find the slope of the given line, we need to rearrange it into the form \( y = mx + c \). Starting from:
\[ x = 3y + 2 \]
we can rearrange it as follows:
\[ 3y = x - 2 \]
\[ y = \frac{1}{3}x - \frac{2}{3} \]

From this, we can see that the slope \( m_1 \) of the line \( x = 3y + 2 \) is:
\[ m_1 = \frac{1}{3} \]

### Step 3: Determine the slope of the line through the origin
Since the two lines are perpendicular, the product of their slopes must equal -1. Let the slope of the line through the origin be \( m \):
\[ m \cdot m_1 = -1 \]
\[ m \cdot \frac{1}{3} = -1 \]
Solving for \( m \):
\[ m = -3 \]

### Step 4: Write the equation of the line through the origin
Now that we have the slope, the equation of the line through the origin is:
\[ y = -3x \]

### Step 5: Solve the two equations simultaneously
We need to find the point of intersection of the lines \( y = -3x \) and \( y = \frac{1}{3}x - \frac{2}{3} \). We can set them equal to each other:
\[ -3x = \frac{1}{3}x - \frac{2}{3} \]

### Step 6: Clear the fractions
To eliminate the fraction, multiply the entire equation by 3:
\[ -9x = x - 2 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ -9x - x = -2 \]
\[ -10x = -2 \]
Dividing by -10:
\[ x = \frac{1}{5} \]

### Step 8: Find the y-coordinate
Now substitute \( x = \frac{1}{5} \) back into the equation of the line through the origin to find \( y \):
\[ y = -3\left(\frac{1}{5}\right) = -\frac{3}{5} \]

### Step 9: Write the coordinates of point X
Thus, the coordinates of point X where the two lines meet at right angles are:
\[ \left( \frac{1}{5}, -\frac{3}{5} \right) \]

### Summary of the solution
The coordinates of point X are:
\[ \left( \frac{1}{5}, -\frac{3}{5} \right) \]