The point (2,3) is a limiting point of a co-axial system of circles of which `x^(2)+y^(2)=9` is a member. The coordinates of the other limiting point is given by

Options

A. `((18)/(13),(27)/(13))`
B. `((9)/(13),(6)/(13))`
C. `((18)/(13)-(27)/(13))`
D. `(-(18)/(13)-(9)/(13))`

Answer

To solve the problem, we need to find the coordinates of the other limiting point of the coaxial system of circles, given that (2, 3) is one of the limiting points and that the circle \(x^2 + y^2 = 9\) is a member of this system.

### Step-by-Step Solution:

1. **Understanding Limiting Points**:
A limiting point of a coaxial system of circles is a point where the circles converge. The radius of the circle at this point is zero.

2. **Equation of the Circle**:
The given circle is \(x^2 + y^2 = 9\). This can be rewritten as:
\[ x^2 + y^2 - 9 = 0 \]

3. **Finding the General Form**:
The general form of the equation for a coaxial system of circles can be expressed as:
\[ x^2 + y^2 - 2\lambda x - 3\lambda y + 11\lambda = 0 \]
where \(\lambda\) is a parameter.

4. **Setting Up the Limiting Point Condition**:
For the limiting point, the radius \(r\) becomes zero. Thus, we set the equation:
\[ \sqrt{(2 - \lambda)^2 + (3 - \frac{3\lambda}{2})^2} = 0 \]
This implies:
\[ (2 - \lambda)^2 + (3 - \frac{3\lambda}{2})^2 = 0 \]

5. **Solving for \(\lambda\)**:
Expanding the above equation:
\[ (2 - \lambda)^2 + (3 - \frac{3\lambda}{2})^2 = 0 \]
This leads to:
\[ (2 - \lambda)^2 + (3 - \frac{3\lambda}{2})^2 = 0 \]
Since both squares must equal zero:
\[ 2 - \lambda = 0 \quad \text{and} \quad 3 - \frac{3\lambda}{2} = 0 \]

6. **Finding Values of \(\lambda\)**:
From \(2 - \lambda = 0\):
\[ \lambda = 2 \]
From \(3 - \frac{3\lambda}{2} = 0\):
\[ 3 = \frac{3\lambda}{2} \implies \lambda = 2 \]

7. **Finding the Other Limiting Point**:
The coordinates of the limiting point are given by:
\[ (\lambda, \frac{3\lambda}{2}) = (2, 3) \]
Now, we need to find the other limiting point. The other limiting point can be found using the relation of the limiting points in a coaxial system, which gives us:
\[ (x_1 + x_2, y_1 + y_2) = (0, 0) \]
where \((x_1, y_1) = (2, 3)\) is the known limiting point.

Therefore, the other limiting point \((x_2, y_2)\) can be calculated as:
\[ x_2 = -2, \quad y_2 = -3 \]

8. **Conclusion**:
The coordinates of the other limiting point are:
\[ (-2, -3) \]