If a circle passes through the point (a,b) and cuts the circles `x^(2)+y^(2)=p^(2)` orthogonally then the equation of locus of its centre is :

Options

A. `x^(2)+y^(2)-2ax -3by +(a^(2)-b^(2)-p^(2))=0`
B. `2ax+2by-(a^(2)+b^(2)+p^(2))=0`
C. `x^(2)+y^(2)-3ax-4by+(a^(2)+b^(2)-p^(2))=0`
D. `2ax+2by -(a^(2)-b^(2)+p^(2))=0`

Answer

To find the locus of the center of a circle that passes through the point (a, b) and cuts the circle \(x^2 + y^2 = p^2\) orthogonally, we can follow these steps:

### Step 1: Define the circles
Let the equation of the circle that we are looking for be:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is its radius.

The given circle is:
\[ x^2 + y^2 = p^2 \]
with center at \((0, 0)\) and radius \(p\).

### Step 2: Orthogonality condition
For two circles to cut orthogonally, the following condition must hold:
\[ d^2 = r_1^2 + r_2^2 \]
where \(d\) is the distance between the centers of the circles, and \(r_1\) and \(r_2\) are their respective radii.

Here, the distance \(d\) between the centers \((h, k)\) and \((0, 0)\) is:
\[ d = \sqrt{h^2 + k^2} \]
Thus, the orthogonality condition becomes:
\[ h^2 + k^2 = r^2 + p^2 \]

### Step 3: Circle passing through (a, b)
Since the circle passes through the point \((a, b)\), we have:
\[ (a - h)^2 + (b - k)^2 = r^2 \]

### Step 4: Substitute \(r^2\)
From the orthogonality condition, we can express \(r^2\) as:
\[ r^2 = h^2 + k^2 - p^2 \]
Substituting this into the equation from Step 3 gives:
\[ (a - h)^2 + (b - k)^2 = h^2 + k^2 - p^2 \]

### Step 5: Expand and simplify
Expanding the left side:
\[ (a^2 - 2ah + h^2) + (b^2 - 2bk + k^2) = h^2 + k^2 - p^2 \]
This simplifies to:
\[ a^2 + b^2 - 2ah - 2bk + h^2 + k^2 = h^2 + k^2 - p^2 \]
Now, cancel \(h^2 + k^2\) from both sides:
\[ a^2 + b^2 - 2ah - 2bk = -p^2 \]
Rearranging gives:
\[ 2ah + 2bk = a^2 + b^2 + p^2 \]

### Step 6: Final equation
Dividing through by 2, we get the locus of the center \((h, k)\):
\[ ah + bk = \frac{a^2 + b^2 + p^2}{2} \]

### Conclusion
The equation of the locus of the center of the circle is:
\[ 2ax + 2by = a^2 + b^2 + p^2 \]