From the point A(0, 3) on the circle `x^(2)+4x+(y-3)^(2)=0`, a chord AB is drawn and extended from point B to a point M such that AM = 2AB. The perimeter of the locus of M is `ppi` units. Then, the value of p is

Answer

To solve the problem step by step, we will follow the outlined approach:

### Step 1: Understand the Circle Equation
The given equation of the circle is:
\[ x^2 + 4x + (y - 3)^2 = 0 \]
We can rewrite this equation to find the center and radius of the circle.

### Step 2: Rewrite the Circle Equation
To rewrite the equation, we complete the square for the \(x\) terms:
\[ x^2 + 4x = (x + 2)^2 - 4 \]
Substituting this back into the circle equation gives:
\[ (x + 2)^2 - 4 + (y - 3)^2 = 0 \]
\[ (x + 2)^2 + (y - 3)^2 = 4 \]
This shows that the circle has a center at \((-2, 3)\) and a radius of \(2\).

### Step 3: Identify Point A and Chord AB
The point \(A(0, 3)\) lies on the circle. We need to draw a chord \(AB\) from point \(A\) and extend it to point \(M\) such that \(AM = 2AB\).

### Step 4: Set Up the Midpoint
Let \(B\) be a point on the circle with coordinates \((h, k)\). Since \(M\) is such that \(AM = 2AB\), point \(B\) will be the midpoint of \(AM\). Thus, using the midpoint formula:
\[ B\left(\frac{0 + h}{2}, \frac{3 + k}{2}\right) = \left(\frac{h}{2}, \frac{3 + k}{2}\right) \]

### Step 5: Substitute B into Circle Equation
Since point \(B\) lies on the circle, we substitute \(\left(\frac{h}{2}, \frac{3 + k}{2}\right)\) into the circle's equation:
\[ \left(\frac{h}{2} + 2\right)^2 + \left(\frac{3 + k}{2} - 3\right)^2 = 4 \]
This simplifies to:
\[ \left(\frac{h + 4}{2}\right)^2 + \left(\frac{k - 3}{2}\right)^2 = 4 \]
Multiplying through by \(4\) gives:
\[ (h + 4)^2 + (k - 3)^2 = 16 \]
This is the equation of a circle centered at \((-4, 3)\) with radius \(4\).

### Step 6: Find the Locus of Point M
Since \(M\) is twice the distance from \(A\) to \(B\), we need to find the locus of \(M\). The coordinates of \(M\) can be expressed in terms of \(h\) and \(k\):
\[ M = (h, k) + 2\left(\frac{h}{2}, \frac{3 + k}{2}\right) = (h + h, k + (3 + k)) = (2h, k + 3) \]
Thus, the locus of point \(M\) can be represented as:
\[ (2h + 4)^2 + (k + 3 - 3)^2 = 16 \]
This simplifies to:
\[ (2h + 4)^2 + k^2 = 16 \]

### Step 7: Calculate the Perimeter of the Locus
The locus of \(M\) is a circle centered at \((-4, 0)\) with a radius of \(4\). The perimeter \(P\) of a circle is given by:
\[ P = 2\pi r = 2\pi \times 4 = 8\pi \]

### Step 8: Identify the Value of p
According to the problem, the perimeter of the locus of \(M\) is given as \(P\pi\). Thus, we compare:
\[ 8\pi = p\pi \]
From this, we can see that \(p = 8\).

### Final Answer
The value of \(p\) is:
\[ \boxed{8} \]