Let `f(x,y)=0` be the equation of a circle. If `f(0,lamda)=0` has equal roots `lamda=2,2 and f(lamda,0)=0` has roots `lamda=(4)/(5),5` then the centre of the circle is

Options

A. `(2,(29)/(10))`
B. `((29)/(10),2)`
C. `(-2,(29)/(10))`
D. None of these

Answer

To find the center of the circle given the conditions in the problem, we can follow these steps:

### Step 1: Assume the equation of the circle
The general equation of a circle can be expressed as:
\[ f(x, y) = x^2 + y^2 + 2gx + 2fy + c = 0 \]

### Step 2: Analyze the first condition
We are given that \( f(0, \lambda) = 0 \) has equal roots \( \lambda = 2, 2 \). Substituting \( x = 0 \) into the circle's equation gives:
\[ f(0, \lambda) = 0^2 + \lambda^2 + 2g(0) + 2f\lambda + c = \lambda^2 + 2f\lambda + c = 0 \]

Since the roots are equal, the discriminant must be zero:
\[ (2f)^2 - 4 \cdot 1 \cdot c = 0 \]
This simplifies to:
\[ 4f^2 - 4c = 0 \]
Thus, we have:
\[ f^2 = c \]

### Step 3: Substitute the roots into the equation
Using the root \( \lambda = 2 \):
\[ 2^2 + 2f(2) + c = 0 \]
This gives:
\[ 4 + 4f + c = 0 \]
Substituting \( c = f^2 \) into this equation:
\[ 4 + 4f + f^2 = 0 \]
Rearranging gives:
\[ f^2 + 4f + 4 = 0 \]
Factoring, we find:
\[ (f + 2)^2 = 0 \]
Thus, \( f = -2 \).

### Step 4: Find \( c \)
Using \( f = -2 \) in \( c = f^2 \):
\[ c = (-2)^2 = 4 \]

### Step 5: Analyze the second condition
Next, we analyze the second condition \( f(\lambda, 0) = 0 \) with roots \( \lambda = \frac{4}{5}, 5 \):
Substituting \( y = 0 \):
\[ f(\lambda, 0) = \lambda^2 + 0^2 + 2g\lambda + 2f(0) + c = \lambda^2 + 2g\lambda + c = 0 \]

Using \( c = 4 \):
\[ \lambda^2 + 2g\lambda + 4 = 0 \]

### Step 6: Use the roots to find \( g \)
The roots are \( \frac{4}{5} \) and \( 5 \). The sum of the roots is:
\[ \frac{4}{5} + 5 = \frac{4}{5} + \frac{25}{5} = \frac{29}{5} \]
Using Vieta's formulas, we know:
\[ -2g = \frac{29}{5} \]
Thus:
\[ g = -\frac{29}{10} \]

### Step 7: Find the center of the circle
The center of the circle is given by the coordinates \( (-g, -f) \):
\[ (-g, -f) = \left(-\left(-\frac{29}{10}\right), -(-2)\right) = \left(\frac{29}{10}, 2\right) \]

### Final Answer
The center of the circle is:
\[ \left(\frac{29}{10}, 2\right) \]