Find the angle through which the axes may be turned so that the equation `Ax+By+C=0` may reduce to the form x = constant, and determine the value of this constant.

Answer

To solve the problem of finding the angle through which the axes may be turned so that the equation \( Ax + By + C = 0 \) reduces to the form \( x = \text{constant} \), we can follow these steps:

### Step 1: Understand the transformation
We need to rotate the coordinate axes by an angle \( \theta \). The transformations for the coordinates are given by:
\[ x' = x \cos \theta - y \sin \theta \]
\[ y' = x \sin \theta + y \cos \theta \]

### Step 2: Substitute the transformations into the equation
We substitute \( x' \) and \( y' \) into the original equation \( Ax + By + C = 0 \):
\[ A(x \cos \theta - y \sin \theta) + B(x \sin \theta + y \cos \theta) + C = 0 \]

### Step 3: Expand and rearrange the equation
Expanding the equation gives:
\[ Ax \cos \theta - Ay \sin \theta + Bx \sin \theta + By \cos \theta + C = 0 \]
Grouping the terms involving \( x \) and \( y \):
\[ (A \cos \theta + B \sin \theta)x + (-A \sin \theta + B \cos \theta)y + C = 0 \]

### Step 4: Set the coefficient of \( y \) to zero
For the equation to reduce to the form \( x = \text{constant} \), the coefficient of \( y \) must be zero:
\[ -A \sin \theta + B \cos \theta = 0 \]

### Step 5: Solve for \( \theta \)
From the equation \( -A \sin \theta + B \cos \theta = 0 \), we can rearrange it to find:
\[ \tan \theta = \frac{B}{A} \]
Thus, the angle \( \theta \) can be expressed as:
\[ \theta = \tan^{-1}\left(\frac{B}{A}\right) \]

### Step 6: Determine the value of the constant
Now, we need to find the value of \( x \) when the equation is in the form \( x = \text{constant} \). We can substitute \( \theta \) back into the original equation:
Using \( \tan \theta = \frac{B}{A} \), we can find the constant \( P \):
\[ x = -\frac{C}{\sqrt{A^2 + B^2}} \]

### Final Result
Thus, the angle through which the axes must be turned is:
\[ \theta = \tan^{-1}\left(\frac{B}{A}\right) \]
And the value of the constant is:
\[ x = -\frac{C}{\sqrt{A^2 + B^2}} \]