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 find the locus of the point M, we start by analyzing the given information and the equation of the circle.

### Step-by-Step Solution:

1. **Identify the Circle Equation**:
The given equation of the circle is:
\[ x^2 + 9x + (y - 3)^2 = 0 \]
We can rewrite this equation by completing the square for the \(x\) terms.

2. **Complete the Square**:
The \(x^2 + 9x\) can be completed as follows:
\[ x^2 + 9x = (x + \frac{9}{2})^2 - \frac{81}{4} \]
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 \(\frac{9}{2}\).

3. **Point A**:
The point A is given as \(A(0, 3)\).

4. **Define Point B**:
Let the coordinates of point B be \(B(x_B, y_B)\). Since B lies on the chord AB, we can express the coordinates of B as a function of M.

5. **Find Point M**:
The point M is defined such that \(AM = 2AB\). This implies that B is the midpoint of A and M. Therefore, if M has coordinates \(M(h, k)\), then the coordinates of B can be expressed as:
\[ B\left(\frac{0 + h}{2}, \frac{3 + k}{2}\right) = \left(\frac{h}{2}, \frac{k + 3}{2}\right) \]

6. **Substitute B into Circle Equation**:
Since point B lies on the circle, we substitute \(B\) into the circle's equation:
\[ \left(\frac{h}{2}\right)^2 + 9\left(\frac{h}{2}\right) + \left(\frac{k + 3}{2} - 3\right)^2 = 0 \]
Simplifying this gives:
\[ \frac{h^2}{4} + \frac{9h}{2} + \left(\frac{k - 3}{2}\right)^2 = 0 \]

7. **Multiply through by 4 to eliminate fractions**:
\[ h^2 + 18h + (k - 3)^2 = 0 \]

8. **Rearranging the Equation**:
This equation can be rearranged to express the locus of point M:
\[ (k - 3)^2 = -h^2 - 18h \]
This shows that the locus of M is a parabola.

9. **Final Locus Equation**:
The locus of M can be expressed in the standard form:
\[ h^2 + 18h + (k - 3)^2 = 0 \]

### Final Answer:
The locus of the point M is given by:
\[ x^2 + 18x + (y - 3)^2 = 0 \]