Origin is a limiting point of a coaxial system of which `x^(2)+y^(2)-6x-8y+1=0` is a member. The other limiting point, is

Options

A. (-2, -4)
B. (3/25, 4/25)
C. (-3/25, -4/25)
D. (4/25, 3/25)

Answer

To solve the problem, we need to find the other limiting point of a coaxial system given that the origin is one of the limiting points. The equation provided is:

\[ x^2 + y^2 - 6x - 8y + 1 = 0 \]

### Step 1: Rewrite the given equation in standard form
We can rewrite the equation in the standard form of a circle by completing the square.

1. Rearranging the equation:
\[ x^2 - 6x + y^2 - 8y + 1 = 0 \]

2. Completing the square for \(x\) and \(y\):
- For \(x^2 - 6x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
- For \(y^2 - 8y\):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

3. Substitute back into the equation:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 + 1 = 0 \]
\[ (x - 3)^2 + (y - 4)^2 - 24 = 0 \]
\[ (x - 3)^2 + (y - 4)^2 = 24 \]

This shows that the center of the circle is at (3, 4) with a radius of \( \sqrt{24} \).

### Step 2: Identify coefficients for the coaxial system
The general form of the equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

From our original equation, we can identify:
- \( g = -3 \)
- \( f = -4 \)
- \( c = 1 \)

### Step 3: Use the formula for the other limiting point
The formula for the other limiting point when one limiting point is at the origin (0, 0) is given by:
\[ \left(-\frac{gc}{g^2 + f^2}, -\frac{fc}{f^2 + g^2}\right) \]

Substituting the values of \( g \), \( f \), and \( c \):
- \( g = -3 \)
- \( f = -4 \)
- \( c = 1 \)

### Step 4: Calculate the coordinates
1. Calculate \( g^2 + f^2 \):
\[ g^2 + f^2 = (-3)^2 + (-4)^2 = 9 + 16 = 25 \]

2. Calculate the x-coordinate:
\[ -\frac{gc}{g^2 + f^2} = -\frac{(-3)(1)}{25} = \frac{3}{25} \]

3. Calculate the y-coordinate:
\[ -\frac{fc}{f^2 + g^2} = -\frac{(-4)(1)}{25} = \frac{4}{25} \]

### Final Result
Thus, the other limiting point is:
\[ \left(\frac{3}{25}, \frac{4}{25}\right) \]