If a circle passes through the point `(3,4) ` and cuts the circle `x ^(2) + y ^(2) =a ^(2)` orthogonally, the equation of the locus of its centre is

Options

A. `3x + 4y -a ^(2) =0`
B. `6x + 8y =a ^(2) +25`
C. `6x + 8y +a ^(2) + 25=0`
D. `3x + 4y =a ^(2) + 25`

Answer

To find the equation of the locus of the center of a circle that passes through the point (3, 4) and cuts the circle \(x^2 + y^2 = a^2\) orthogonally, we can follow these steps:

### Step 1: General Equation of the Circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where the center of the circle is at the point \((-g, -f)\).

### Step 2: Circle Passing Through (3, 4)
Since the circle passes through the point (3, 4), we can substitute these coordinates into the general equation:
\[ 3^2 + 4^2 + 2g(3) + 2f(4) + c = 0 \]
Calculating \(3^2 + 4^2\):
\[ 9 + 16 + 6g + 8f + c = 0 \]
This simplifies to:
\[ 25 + 6g + 8f + c = 0 \quad \text{(1)} \]

### Step 3: Condition for Orthogonality
For the two circles to intersect orthogonally, the following condition must hold:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
For the circle \(x^2 + y^2 = a^2\), we can rewrite it as:
\[ x^2 + y^2 + 0x + 0y - a^2 = 0 \]
Here, \(g_2 = 0\), \(f_2 = 0\), and \(c_2 = -a^2\).

Substituting into the orthogonality condition:
\[ 2g(0) + 2f(0) = c + (-a^2) \]
This simplifies to:
\[ 0 = c - a^2 \quad \Rightarrow \quad c = a^2 \quad \text{(2)} \]

### Step 4: Substitute \(c\) into Equation (1)
Now, substituting \(c = a^2\) into equation (1):
\[ 25 + 6g + 8f + a^2 = 0 \]
Rearranging gives:
\[ 6g + 8f = -25 - a^2 \quad \text{(3)} \]

### Step 5: Locus of the Center
The center of the circle is at the point \((-g, -f)\). Replacing \(g\) with \(-x\) and \(f\) with \(-y\) (where \((x, y)\) is the center of the circle), we have:
\[ 6(-x) + 8(-y) = -25 - a^2 \]
This simplifies to:
\[ -6x - 8y = -25 - a^2 \]
Multiplying through by -1:
\[ 6x + 8y = 25 + a^2 \]

### Final Step: Locus Equation
Thus, the equation of the locus of the center of the circle is:
\[ 6x + 8y = a^2 + 25 \]