If the origin is shifted to the point `((a b)/(a-b),0)` without rotation, then the equation `(a-b)(x^2+y^2)-2a b x=0` becomes

Options

A. `(a-b)(x^2+y^2)-(a+b)xy+abx=a^2`
B. `(a+b)(x^2+y^2)=2ab`
C. `(x^2+y^2)=(a^2+b^2)`
D. `(a-b)^2(x^2+y^2)=a^2b^2`

Explanation

To solve the problem, we need to shift the origin from (0, 0) to the point \(\left(\frac{a b}{a - b}, 0\right)\) and find the new equation of the given equation \((a - b)(x^2 + y^2) - 2abx = 0\).

### Step-by-Step Solution:

1. **Identify the given equation:**
The original equation is:
\[ (a - b)(x^2 + y^2) - 2abx = 0 \]

2. **Shift the origin:**
When the origin is shifted to \(\left(\frac{ab}{a - b}, 0\right)\), we need to express the new coordinates \(X\) and \(Y\) in terms of the old coordinates \(x\) and \(y\):
\[ x = X + \frac{ab}{a - b}, \quad y = Y \]

3. **Substitute the new coordinates into the equation:**
Substitute \(x\) and \(y\) into the original equation:
\[ (a - b)\left(X + \frac{ab}{a - b}\right)^2 + Y^2 - 2ab\left(X + \frac{ab}{a - b}\right) = 0 \]

4. **Expand the equation:**
Expanding \(\left(X + \frac{ab}{a - b}\right)^2\):
\[ = X^2 + 2X\frac{ab}{a - b} + \left(\frac{ab}{a - b}\right)^2 \]
Thus, the equation becomes:
\[ (a - b)\left(X^2 + 2X\frac{ab}{a - b} + \frac{a^2b^2}{(a - b)^2}\right) + Y^2 - 2ab\left(X + \frac{ab}{a - b}\right) = 0 \]

5. **Simplify the equation:**
Distributing \((a - b)\):
\[ (a - b)X^2 + 2abX + \frac{a^2b^2}{a - b} + Y^2 - 2abX - \frac{2a^2b}{a - b} = 0 \]
The \(2abX\) terms cancel out:
\[ (a - b)X^2 + Y^2 + \frac{a^2b^2 - 2a^2b}{a - b} = 0 \]
Simplifying further:
\[ (a - b)X^2 + Y^2 + \frac{ab(a - b)}{a - b} = 0 \]
This leads to:
\[ (a - b)X^2 + Y^2 + ab = 0 \]

6. **Final equation:**
Rearranging gives us:
\[ (a - b)(X^2 + Y^2) = a^2b^2 \]

### Conclusion:
The final equation after shifting the origin is:
\[ (a - b)(X^2 + Y^2) = a^2b^2 \]