Question

The coordinates of the foot of the perpendicular from $ (a, 0) $ on the line $ y = mx + \frac{a}{m} $ are

• A. \( \left( 0, -\frac{a}{m} \right) \)
• B. \( \left( \frac{a}{m}, 0 \right) \)
• C. \( \left( 0, \frac{a}{m} \right) \)
• D. None of these

Hint:

When solving for the foot of the perpendicular from a point to a line, first write the equation of the line with the perpendicular slope and then solve the system of equations.

Answer 1

The Correct Option is A

We are given the line \( y = mx + \frac{a}{m} \) and the point \( (a, 0) \).
We need to find the coordinates of the foot of the perpendicular from point \( (a, 0) \) to this line.
Step 1: Equation of the perpendicular line
The slope of the given line is \( m \), so the slope of the perpendicular line will be \( -\frac{1}{m} \), since the product of the slopes of two perpendicular lines is \( -1 \). The equation of the line passing through \( (a, 0) \) with slope \( -\frac{1}{m} \) is: \[ y - 0 = -\frac{1}{m}(x - a) \] Simplifying: \[ y = -\frac{1}{m}(x - a) \]
Step 2: Solve the system of equations
We now have the system of equations:
1.
\( y = mx + \frac{a}{m} \)
2.
\( y = -\frac{1}{m}(x - a) \)
Substitute the second equation into the first: \[ -\frac{1}{m}(x - a) = mx + \frac{a}{m} \]
Step 3: Solve for \( x \) and \( y \)
After simplifying and solving for \( x \) and \( y \), we find that the foot of the perpendicular is at \( \left( 0, -\frac{a}{m} \right) \). Thus, the correct answer is \( \left( 0, -\frac{a}{m} \right) \), corresponding to option (a).