The focus of rectangular hyperbola `(x-a)*(y-b)=c^2` is

Options

A. (a) `(h-p, k-p)`
B. (b) `(h-p, k+p)`
C. (c) `(h+p, k-p)`
D. (d) None of these

Answer

To find the focus of the rectangular hyperbola given by the equation \((x-a)(y-b) = c^2\), we can follow these steps:

### Step 1: Understand the Standard Form
The standard form of a rectangular hyperbola is given by \(xy = c^2\). The equation \((x-a)(y-b) = c^2\) can be transformed into this standard form.

### Step 2: Expand the Given Equation
We can expand the equation:
\[ (x-a)(y-b) = xy - by - ax + ab = c^2 \]
This gives us:
\[ xy - ax - by + ab - c^2 = 0 \]

### Step 3: Identify the Center
The center of the hyperbola can be found by observing that the equation can be rewritten as:
\[ xy = c^2 + ax + by - ab \]
The center of the hyperbola is at the point \((a, b)\).

### Step 4: Find the Foci
For a rectangular hyperbola of the form \(xy = c^2\), the foci are located at:
\[ (\pm \sqrt{2}c, 0) \text{ and } (0, \pm \sqrt{2}c) \]
Since we have translated the hyperbola by \((a, b)\), we can find the foci by adding \((a, b)\) to the coordinates of the foci.

### Step 5: Calculate the Coordinates of the Foci
Thus, the coordinates of the foci will be:
1. \((a + \sqrt{2}c, b + \sqrt{2}c)\)
2. \((a - \sqrt{2}c, b - \sqrt{2}c)\)
3. \((a + \sqrt{2}c, b - \sqrt{2}c)\)
4. \((a - \sqrt{2}c, b + \sqrt{2}c)\)

### Final Result
The foci of the rectangular hyperbola \((x-a)(y-b) = c^2\) are:
\[ (a + \sqrt{2}c, b + \sqrt{2}c), (a - \sqrt{2}c, b - \sqrt{2}c), (a + \sqrt{2}c, b - \sqrt{2}c), (a - \sqrt{2}c, b + \sqrt{2}c) \]