From the point A(0, 3) on the circle `x^(2)+9x+(y-3)^(2)=0,` a chord AB is drawn and extended to a point M such that AM = 2AB (B lies between A & M). The locus of the point M is

Options

A. `x^(2)+18x+y^(2)=0`
B. `x^(2)+18x+(y-3)^(2)=0`
C. `(x-3)^(2)+18x+y^(2)=0`
D. `x^(2)+8x+18-y^(2)=0`

Answer

To solve the problem step by step, we will find the locus of point M given the conditions of the problem.

### Step 1: Understand the Circle Equation
The equation of the circle is given as:
\[ x^2 + 9x + (y - 3)^2 = 0 \]
We can rewrite this equation in standard form by completing the square.

### Step 2: Completing the Square
1. For \(x^2 + 9x\), we complete the square:
\[ x^2 + 9x = (x + \frac{9}{2})^2 - \frac{81}{4} \]
2. Thus, the equation becomes:
\[ (x + \frac{9}{2})^2 + (y - 3)^2 = \frac{81}{4} \]
This represents a circle centered at \((-4.5, 3)\) with a radius of \(\frac{9}{2}\).

### Step 3: Identify Point A
The point A is given as \(A(0, 3)\). We can confirm that this point lies on the circle by substituting \(x = 0\) and \(y = 3\) into the circle's equation:
\[ 0^2 + 9(0) + (3 - 3)^2 = 0 \quad \text{(True)} \]

### Step 4: Define Point B and Point M
Let the coordinates of point B be \((x_1, y_1)\). The problem states that \(AM = 2AB\) and \(B\) lies between \(A\) and \(M\). This means:
\[ AM = AB + MB \quad \text{and} \quad AM = 2AB \implies MB = AB \]
Thus, point B divides the segment \(AM\) in the ratio \(1:1\).

### Step 5: Use Section Formula
Using the section formula, the coordinates of point B can be expressed as:
\[ x_1 = \frac{h + 0}{2} = \frac{h}{2} \]
\[ y_1 = \frac{k + 3}{2} \]
where \(M(h, k)\).

### Step 6: Substitute B into Circle Equation
Since point B lies on the circle, we substitute \(x_1\) and \(y_1\) into the circle's equation:
\[ \left(\frac{h}{2}\right)^2 + 9\left(\frac{h}{2}\right) + \left(\frac{k + 3}{2} - 3\right)^2 = 0 \]
This simplifies to:
\[ \frac{h^2}{4} + \frac{9h}{2} + \left(\frac{k - 3}{2}\right)^2 = 0 \]

### Step 7: Simplify the Equation
1. Multiply through by 4 to eliminate the fractions:
\[ h^2 + 18h + (k - 3)^2 = 0 \]
2. Rearranging gives:
\[ h^2 + 18h + (k - 3)^2 = 0 \]

### Step 8: Replace Variables
Replace \(h\) with \(x\) and \(k\) with \(y\):
\[ x^2 + 18x + (y - 3)^2 = 0 \]

### Step 9: Final Result
Thus, the locus of point M is given by:
\[ x^2 + 18x + (y - 3)^2 = 0 \]