Finding the Foot of the Perpendicular from a Point to a Line

The problem asks us to find the coordinates of the foot of the perpendicular drawn from the point A(2, 4) to the line L given by the equation \(x + y = 1\).

Understanding the Concept

The foot of the perpendicular is the point of intersection between the given line and a new line that passes through the given point and is perpendicular to the given line.

Step 1: Find the slope of the given line

The equation of the given line L is \(x + y = 1\). We can rewrite this in the slope-intercept form (\(y = mx + c\)) to find the slope:

\(y = -x + 1\)

The slope of line L, denoted by \(m_L\), is -1.

Step 2: Find the slope of the perpendicular line

Let the slope of the perpendicular line be \(m_{perp}\). For two lines to be perpendicular, the product of their slopes must be -1.

\(m_L \times m_{perp} = -1\)

\((-1) \times m_{perp} = -1\)

\(m_{perp} = \frac{-1}{-1} = 1\)

So, the slope of the line perpendicular to \(x + y = 1\) is 1.

Step 3: Find the equation of the perpendicular line

This perpendicular line passes through the point A(2, 4) and has a slope \(m_{perp} = 1\). Using the point-slope form of a line equation (\(y - y_1 = m(x - x_1)\)):

\(y - 4 = 1(x - 2)\)

\(y - 4 = x - 2\)

Rearranging the terms, we get the equation of the perpendicular line:

\(y = x + 2\)

Step 4: Find the intersection point (Foot of the perpendicular)

The foot of the perpendicular is the point where the original line (\(x + y = 1\)) and the perpendicular line (\(y = x + 2\)) intersect. We can solve these two equations simultaneously:

1. \(x + y = 1\)
2. \(y = x + 2\)

Substitute the expression for \(y\) from equation (2) into equation (1):

\(x + (x + 2) = 1\)

\(2x + 2 = 1\)

\(2x = 1 - 2\)

\(2x = -1\)

\(x = -\frac{1}{2}\)

Now substitute the value of \(x\) back into equation (2) to find \(y\):

\(y = x + 2\)

\(y = -\frac{1}{2} + 2\)

\(y = -\frac{1}{2} + \frac{4}{2}\)

\(y = \frac{3}{2}\)

Conclusion

The coordinates of the foot of the perpendicular from the point (2, 4) upon the line \(x + y = 1\) are \(\left(-\frac{1}{2}, \frac{3}{2}\right)\).