Find the foot of the perpendicular from the point `(-1,2)` on the st. Line `x-y+5=0`.

Answer

To find the foot of the perpendicular from the point \((-1, 2)\) to the line given by the equation \(x - y + 5 = 0\), we can follow these steps:

### Step 1: Identify the coefficients of the line equation
The line equation can be rewritten in the standard form \(Ax + By + C = 0\). From the equation \(x - y + 5 = 0\), we identify:
- \(A = 1\)
- \(B = -1\)
- \(C = 5\)

### Step 2: Use the formula for the foot of the perpendicular
The coordinates of the foot of the perpendicular from point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) can be found using the formula:
\[ \left( x, y \right) = \left( x_1 - \frac{A(Ax_1 + By_1 + C)}{A^2 + B^2}, y_1 - \frac{B(Ax_1 + By_1 + C)}{A^2 + B^2} \right) \]
Here, \((x_1, y_1) = (-1, 2)\).

### Step 3: Calculate \(Ax_1 + By_1 + C\)
Substituting \(x_1 = -1\) and \(y_1 = 2\):
\[ Ax_1 + By_1 + C = 1(-1) + (-1)(2) + 5 = -1 - 2 + 5 = 2 \]

### Step 4: Calculate \(A^2 + B^2\)
Now, we calculate:
\[ A^2 + B^2 = 1^2 + (-1)^2 = 1 + 1 = 2 \]

### Step 5: Substitute into the formula
Now we substitute the values into the formula:
\[ x = -1 - \frac{1 \cdot 2}{2} = -1 - 1 = -2 \]
\[ y = 2 - \frac{-1 \cdot 2}{2} = 2 + 1 = 3 \]

### Step 6: Write the final answer
The foot of the perpendicular from the point \((-1, 2)\) to the line \(x - y + 5 = 0\) is \((-2, 3)\).

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