What are the coordinates of the foot of the perpendicular from the point (2 , 3) on the line x + y - 11 = 0 ?

Options

A. (2 , 9)
B. (5 , 6)
C. (-5 , 6)
D. (6,5)

Answer

To find the coordinates of the foot of the perpendicular from the point (2, 3) to the line \(x + y - 11 = 0\), we can follow these steps:

### Step 1: Identify the given point and the line equation
The given point is \(P(2, 3)\) and the line equation is \(x + y - 11 = 0\).

### Step 2: Find the slope of the line
To find the slope of the line \(x + y - 11 = 0\), we can rewrite it in slope-intercept form \(y = mx + b\):
\[ y = -x + 11 \]
From this, we see that the slope \(m_1\) of the line is \(-1\).

### Step 3: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope \(m_2\) of the perpendicular line is:
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-1} = 1 \]

### Step 4: Write the equation of the perpendicular line
Using the point-slope form of the equation of a line, the equation of the line passing through point \(P(2, 3)\) with slope \(1\) is:
\[ y - 3 = 1(x - 2) \]
Simplifying this, we get:
\[ y - 3 = x - 2 \implies y = x + 1 \]

### Step 5: Find the intersection of the two lines
Now, we need to find the intersection of the lines \(y = x + 1\) and \(x + y - 11 = 0\). Substituting \(y\) from the second equation into the first:
\[ x + (x + 1) - 11 = 0 \]
This simplifies to:
\[ 2x + 1 - 11 = 0 \implies 2x - 10 = 0 \implies 2x = 10 \implies x = 5 \]

### Step 6: Find the corresponding y-coordinate
Now, substituting \(x = 5\) back into the equation \(y = x + 1\):
\[ y = 5 + 1 = 6 \]

### Conclusion
Thus, the coordinates of the foot of the perpendicular from the point (2, 3) to the line \(x + y - 11 = 0\) are \((5, 6)\).

### Final Answer
The coordinates of the foot of the perpendicular are \((5, 6)\).

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