The coordinates of the foot of the perpendicular from (a,0) on the line `y = mx + a/m ` are

Options

A. `(0,-1/a)`
B. `(0,a/m)`
C. `(0,-a/m)`
D. `(0,1/a)`

Answer

To find the coordinates of the foot of the perpendicular from the point \( (a, 0) \) to the line given by the equation \( y = mx + \frac{a}{m} \), we can follow these steps:

### Step 1: Convert the line equation to standard form
The given line is in slope-intercept form \( y = mx + \frac{a}{m} \). We need to convert it into the standard form \( Ax + By + C = 0 \).

\[ y - mx - \frac{a}{m} = 0 \implies mx - y + \frac{a}{m} = 0 \]

Here, we can identify:
- \( A = m \)
- \( B = -1 \)
- \( C = \frac{a}{m} \)

### Step 2: Use the formula for the foot of the perpendicular
The coordinates of the foot of the perpendicular from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) are given by the formula:

\[ \left( \frac{B(Bx_1 - Ay_1) - AC}{A^2 + B^2}, \frac{A(Ay_1 - Bx_1) - BC}{A^2 + B^2} \right) \]

In our case, \( (x_1, y_1) = (a, 0) \).

### Step 3: Substitute the values into the formula
Substituting \( A = m \), \( B = -1 \), \( C = \frac{a}{m} \), \( x_1 = a \), and \( y_1 = 0 \):

1. Calculate the x-coordinate:
\[ x_2 = \frac{-1(-1 \cdot a - m \cdot 0) - m \cdot \frac{a}{m}}{m^2 + (-1)^2} \]
\[ = \frac{a - a}{m^2 + 1} = \frac{0}{m^2 + 1} = 0 \]

2. Calculate the y-coordinate:
\[ y_2 = \frac{m(0 - (-1 \cdot a)) - (-1) \cdot \frac{a}{m}}{m^2 + 1} \]
\[ = \frac{ma + \frac{a}{m}}{m^2 + 1} \]
\[ = \frac{a(m + \frac{1}{m})}{m^2 + 1} = \frac{a \cdot \frac{m^2 + 1}{m}}{m^2 + 1} = \frac{a}{m} \]

### Step 4: Write the final coordinates
Thus, the coordinates of the foot of the perpendicular from the point \( (a, 0) \) to the line \( y = mx + \frac{a}{m} \) are:

\[ (0, \frac{a}{m}) \]

### Conclusion
The final answer is \( (0, \frac{a}{m}) \).