Reduce the equation in the Normal form `-x+y+4=0` . Find its perpendicular distance from the origin and angle between perpendicular and the positive x- axis .

Answer

To solve the problem step by step, we will first reduce the equation \(-x + y + 4 = 0\) to its normal form, then find the perpendicular distance from the origin, and finally calculate the angle between the perpendicular and the positive x-axis.

### Step 1: Rearranging the Equation
We start with the equation:
\[ -x + y + 4 = 0 \]
Rearranging it gives:
\[ y = x - 4 \]

### Step 2: Writing in Standard Form
We can rewrite the equation in the standard form \(Ax + By + C = 0\):
\[ x - y + 4 = 0 \]
Here, \(A = 1\), \(B = -1\), and \(C = 4\).

### Step 3: Finding the Normal Form
The normal form of a line is given by:
\[ \frac{x}{\sqrt{A^2 + B^2}} + \frac{y}{\sqrt{A^2 + B^2}} = \frac{-C}{\sqrt{A^2 + B^2}} \]
Calculating \(A^2 + B^2\):
\[ A^2 + B^2 = 1^2 + (-1)^2 = 1 + 1 = 2 \]
Thus, \(\sqrt{A^2 + B^2} = \sqrt{2}\).

Now substituting into the normal form:
\[ \frac{x}{\sqrt{2}} - \frac{y}{\sqrt{2}} = -\frac{4}{\sqrt{2}} \]
This simplifies to:
\[ \frac{x}{\sqrt{2}} - \frac{y}{\sqrt{2}} = -2\sqrt{2} \]

### Step 4: Finding the Perpendicular Distance from the Origin
The perpendicular distance \(P\) from the origin to the line \(Ax + By + C = 0\) is given by:
\[ P = \frac{|C|}{\sqrt{A^2 + B^2}} \]
Substituting the values:
\[ P = \frac{|4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Step 5: Finding the Angle with the Positive X-axis
The angle \(\alpha\) between the line and the positive x-axis can be found using the coefficients \(A\) and \(B\):
\[ \tan(\alpha) = \frac{B}{A} = \frac{-1}{1} = -1 \]
Thus, \(\alpha = \tan^{-1}(-1)\). The angle corresponding to \(-1\) in the fourth quadrant is:
\[ \alpha = 360^\circ - 45^\circ = 315^\circ \]

### Final Answers
- The normal form of the equation is:
\[ \frac{x}{\sqrt{2}} - \frac{y}{\sqrt{2}} = -2\sqrt{2} \]
- The perpendicular distance from the origin is:
\[ 2\sqrt{2} \]
- The angle between the perpendicular and the positive x-axis is:
\[ 315^\circ \]