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: Write the equation of the line in standard form
The given line is \( y = mx + \frac{a}{m} \). We can rearrange this into the standard form \( Ax + By + C = 0 \).

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

Here, \( A = m \), \( B = -1 \), and \( C = \frac{a}{m} \).

### 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 \( Ax + By + C = 0 \) can be found using the formula:

\[ \left( x, y \right) = \left( x_1 - \frac{A(Ax_1 + By_1 + C)}{A^2 + B^2}, y_1 - \frac{B(Ax_1 + By_1 + C)}{A^2 + B^2} \right) \]

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

### Step 3: Calculate \( Ax_1 + By_1 + C \)
Substituting \( A \), \( B \), \( C \), \( x_1 \), and \( y_1 \):

\[ Ax_1 + By_1 + C = m(a) + (-1)(0) + \frac{a}{m} = ma + \frac{a}{m} \]

### Step 4: Calculate \( A^2 + B^2 \)
\[ A^2 + B^2 = m^2 + (-1)^2 = m^2 + 1 \]

### Step 5: Substitute into the foot of the perpendicular formula
Now substitute these values into the formula:

\[ x = a - \frac{m(ma + \frac{a}{m})}{m^2 + 1} \]
\[ y = 0 - \frac{-1(ma + \frac{a}{m})}{m^2 + 1} \]

### Step 6: Simplify the expressions for \( x \) and \( y \)
**For \( x \):**
\[ x = a - \frac{m^2a + a}{m^2 + 1} = a - \frac{a(m^2 + 1)}{m^2 + 1} = a - a = 0 \]

**For \( y \):**
\[ y = \frac{ma + \frac{a}{m}}{m^2 + 1} = \frac{a(m + \frac{1}{m})}{m^2 + 1} = \frac{a\left(\frac{m^2 + 1}{m}\right)}{m^2 + 1} = \frac{a}{m} \]

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

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