The coordinates of the foot of the perpendicular from the point (2,3) on the line `x+y-11=0` are (i) (-6,5) (ii) (5,6) (iii) (-5,6) (iv) (6,5)

Options

A. (-6,5)
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 given by the equation \(x + y - 11 = 0\), we can follow these steps:

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

### Step 2: Find the slope of the line
The equation of the line can be rewritten in slope-intercept form \(y = -x + 11\). From this, we can see that the slope of the line is \(-1\).

### Step 3: Determine the slope of the perpendicular line
The slope of the line perpendicular to this line will be the negative reciprocal of \(-1\), which is \(1\).

### 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:
\[ y - 3 = 1(x - 2) \]
This simplifies to:
\[ y = x + 1 \]

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

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

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

### Conclusion
The correct answer is option (ii) \((5, 6)\).

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