Find the foot of `bot` of a point (2,3) to line `x+y-11=0`

Answer

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

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

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

### Step 3: Determine the slope of the perpendicular line
The slope of the perpendicular line is the negative reciprocal of the slope of the original line. Therefore, if the slope of the line is \(-1\), the slope of the perpendicular line is:
\[ m_{\text{perpendicular}} = 1 \]

### Step 4: Write the equation of the perpendicular line
Using the point-slope form of the equation of a line, the equation of the perpendicular line passing through point \(P(2, 3)\) can be written as:
\[ y - 3 = 1(x - 2) \]
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 line \(y = x + 1\) and the line \(x + y - 11 = 0\). We can substitute \(y\) from the first equation into the second:
\[ x + (x + 1) - 11 = 0 \]
This simplifies to:
\[ 2x + 1 - 11 = 0 \implies 2x - 10 = 0 \implies 2x = 10 \implies x = 5 \]

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

### Step 7: Write the coordinates of the foot of the perpendicular
Thus, the foot of the perpendicular from the point (2, 3) to the line \(x + y - 11 = 0\) is:
\[ (5, 6) \]

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

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