If `4x+3y-12=0` touches `(x-p)^(2)+(y-p)^(2)=p^(2)`, then the sum of all the possible values of p is

Answer

To solve the problem, we need to find the sum of all possible values of \( p \) such that the line \( 4x + 3y - 12 = 0 \) touches the circle defined by \( (x - p)^2 + (y - p)^2 = p^2 \).

### Step-by-Step Solution:

1. **Identify the Center and Radius of the Circle:**
The equation of the circle is given by:
\[ (x - p)^2 + (y - p)^2 = p^2 \]
From this, we can see that the center of the circle is \( (p, p) \) and the radius is \( p \).

2. **Distance from the Center to the Line:**
The distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
For our line \( 4x + 3y - 12 = 0 \), we have \( A = 4 \), \( B = 3 \), and \( C = -12 \). The center of the circle is \( (p, p) \).

Plugging in the values:
\[ d = \frac{|4p + 3p - 12|}{\sqrt{4^2 + 3^2}} = \frac{|7p - 12|}{\sqrt{25}} = \frac{|7p - 12|}{5} \]

3. **Setting the Distance Equal to the Radius:**
Since the line touches the circle, the distance from the center to the line must equal the radius of the circle:
\[ \frac{|7p - 12|}{5} = p \]

4. **Solving the Equation:**
We can multiply both sides by 5 to eliminate the fraction:
\[ |7p - 12| = 5p \]

This absolute value equation gives us two cases to consider:

**Case 1:**
\[ 7p - 12 = 5p \]
Solving this:
\[ 7p - 5p = 12 \implies 2p = 12 \implies p = 6 \]

**Case 2:**
\[ 7p - 12 = -5p \]
Solving this:
\[ 7p + 5p = 12 \implies 12p = 12 \implies p = 1 \]

5. **Finding the Sum of All Possible Values of \( p \):**
The possible values of \( p \) are \( 6 \) and \( 1 \). Therefore, the sum of all possible values of \( p \) is:
\[ 6 + 1 = 7 \]

### Final Answer:
The sum of all possible values of \( p \) is \( 7 \).