If `Ax+By=C and x"cos"alpha+y"sin"alpha=p` represent the same line, find p in terms of A, B, C.

Answer

To solve the problem, we need to find \( p \) in terms of \( A \), \( B \), and \( C \) given that the equations \( Ax + By = C \) and \( x \cos \alpha + y \sin \alpha = p \) represent the same line.

### Step-by-Step Solution:

1. **Identify the equations**: We have two equations:
\[ Ax + By = C \quad \text{(1)} \]
\[ x \cos \alpha + y \sin \alpha = p \quad \text{(2)} \]

2. **Set up the ratio of coefficients**: Since both equations represent the same line, the ratios of the coefficients of \( x \), \( y \), and the constants must be equal. Therefore, we can write:
\[ \frac{A}{\cos \alpha} = \frac{B}{\sin \alpha} = \frac{C}{p} \quad \text{(3)} \]

3. **Express the ratios**: From equation (3), we can express \( p \) in terms of \( A \), \( B \), and \( C \). Let's denote the common ratio as \( k \):
\[ A = k \cos \alpha \quad \text{(4)} \]
\[ B = k \sin \alpha \quad \text{(5)} \]
\[ C = kp \quad \text{(6)} \]

4. **Substitute for \( k \)**: From equations (4) and (5), we can express \( k \):
\[ k = \frac{A}{\cos \alpha} \quad \text{or} \quad k = \frac{B}{\sin \alpha} \]

5. **Equate and solve for \( p \)**: From equation (6), we can substitute \( k \):
\[ C = \frac{A}{\cos \alpha} p \quad \text{or} \quad C = \frac{B}{\sin \alpha} p \]
Rearranging gives:
\[ p = \frac{C \cos \alpha}{A} \quad \text{(7)} \]
or
\[ p = \frac{C \sin \alpha}{B} \quad \text{(8)} \]

6. **Use the Pythagorean identity**: Since \( \cos^2 \alpha + \sin^2 \alpha = 1 \), we can relate \( p \) to \( A \), \( B \), and \( C \) using the Pythagorean identity:
\[ p^2 = \frac{C^2}{\frac{A^2}{\cos^2 \alpha} + \frac{B^2}{\sin^2 \alpha}} \quad \text{(9)} \]

7. **Final expression for \( p \)**: We can simplify this to:
\[ p = \frac{C}{\sqrt{A^2 + B^2}} \quad \text{(10)} \]

### Final Result:
Thus, the value of \( p \) in terms of \( A \), \( B \), and \( C \) is:
\[ p = \frac{C}{\sqrt{A^2 + B^2}} \]