The locus of the centres of circles passing through the origin and intersecting the fixed circle `x^(2)+y^(2)-5x+3y-1=0` orthogonally is

Options

A. a straight line of slope 3/5
B. a circle
C. a pair of straight lines
D. none of these

Explanation

To find the locus of the centers of circles passing through the origin and intersecting the fixed circle orthogonally, we can follow these steps:

### Step 1: Write the general equation of the circle
Let the equation of the circle passing through the origin be given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Since the circle passes through the origin (0, 0), substituting \(x = 0\) and \(y = 0\) gives:
\[ 0 + 0 + 0 + 0 + c = 0 \implies c = 0 \]
Thus, the equation simplifies to:
\[ x^2 + y^2 + 2gx + 2fy = 0 \]

### Step 2: Identify the center of the circle
The center of this circle is given by the coordinates:
\[ (-g, -f) \]

### Step 3: Use the condition for orthogonal intersection
For two circles to intersect orthogonally, the condition is:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
where \(g_1, f_1, c_1\) are the parameters of the first circle and \(g_2, f_2, c_2\) are for the second circle.

### Step 4: Identify the fixed circle
The fixed circle is given by:
\[ x^2 + y^2 - 5x + 3y - 1 = 0 \]
Comparing with the general form, we have:
- \(g_2 = -\frac{5}{2}\)
- \(f_2 = \frac{3}{2}\)
- \(c_2 = -1\)

### Step 5: Substitute values into the orthogonality condition
For our circle, we have:
- \(g_1 = g\)
- \(f_1 = f\)
- \(c_1 = 0\)

Substituting into the orthogonality condition:
\[ 2g(-\frac{5}{2}) + 2f(\frac{3}{2}) = 0 + (-1) \]
This simplifies to:
\[ -5g + 3f = -1 \]

### Step 6: Express in terms of x and y
Since \(g = -x\) and \(f = -y\) (where \((-g, -f)\) are the coordinates of the center), we substitute:
\[ -5(-x) + 3(-y) = -1 \]
This simplifies to:
\[ 5x - 3y = -1 \]

### Step 7: Rearrange to find the locus
Rearranging gives us the equation of the locus:
\[ 5x - 3y + 1 = 0 \]

### Conclusion
The locus of the centers of circles passing through the origin and intersecting the fixed circle orthogonally is:
\[ 5x - 3y + 1 = 0 \]