Question

The straight line passing through \( (0, 0) \) and the foot of the perpendicular from \( (2, 4) \) onto the line \( x + y - 1 = 0 \) is:

• A. \( y = -3x \)
• B. \( y = 3x \)
• C. \( y = \frac{1}{3}x \)
• D. \( y = -\frac{1}{3}x \)

Hint:

Foot of Perpendicular and Line Equation}
Use foot of perpendicular formula to avoid solving systems
Line from origin with known point uses slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Once you have slope, form equation as \( y = mx \)

Answer 1

The Correct Option is A

We are to find the line through origin and the foot of perpendicular from \( (2, 4) \) to the line \( x + y - 1 = 0 \). Step 1: Find slope of the given line Given line: \( x + y = 1 \Rightarrow y = -x + 1 \Rightarrow m_1 = -1 \) So, the slope of the perpendicular from point to line is: \[ m_2 = \frac{1}{m_1} = \frac{1}{-1} = -1 \] But we will use the formula to directly find foot of perpendicular from \( (x_1, y_1) \) to \( ax + by + c = 0 \): Foot of perpendicular from \( (x_1, y_1) \) to line \( ax + by + c = 0 \) is: \[ \left( x - a \cdot \frac{ax + by + c}{a^2 + b^2}, y - b \cdot \frac{ax + by + c}{a^2 + b^2} \right) \] Here: \[ (x_1, y_1) = (2, 4),\quad a = 1,\quad b = 1,\quad c = -1 \Rightarrow \text{Foot} = \left( 2 - \frac{(2 + 4 - 1)}{2}, 4 - \frac{(2 + 4 - 1)}{2} \right) = (2 - \frac{5}{2}, 4 - \frac{5}{2}) = \left( -\frac{1}{2}, \frac{3}{2} \right) \] Now slope of line joining \( (0, 0) \) and \( \left( -\frac{1}{2}, \frac{3}{2} \right) \) is: \[ m = \frac{3/2 - 0}{-1/2 - 0} = \frac{3}{-1} = -3 \Rightarrow y = -3x \]