The equation of the chord joining the points `(x_(1) , y_(1))` and `(x_(2), y_(2))` on the rectangular hyperbola xy = `c^(2)` is :

Options

A. `(x)/(x_(1) + x_(2)) + (y)/(y_(1) + y_(2))` = 1
B. `(x)/(x_(1) - x_(2)) + (y)/(y_(1) - y_(2))` = 1
C. `(x)/(y_(1) + y_(2)) + (y)/(x_(1) + x_(2))` = 1
D. `(x)/(y_(1) - y_(2)) + (y)/(x_(1) - x_(2))` = 1

Answer

To find the equation of the chord joining the points \((x_1, y_1)\) and \((x_2, y_2)\) on the rectangular hyperbola given by \(xy = c^2\), we can follow these steps:

### Step 1: Verify that the points lie on the hyperbola
Since the points \((x_1, y_1)\) and \((x_2, y_2)\) lie on the hyperbola \(xy = c^2\), we can write:
\[ y_1 = \frac{c^2}{x_1} \quad \text{and} \quad y_2 = \frac{c^2}{x_2} \]

### Step 2: Find the slope of the chord
The slope \(m\) of the line joining the two points can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values of \(y_1\) and \(y_2\):
\[ m = \frac{\frac{c^2}{x_2} - \frac{c^2}{x_1}}{x_2 - x_1} \]
To simplify this, we find a common denominator:
\[ m = \frac{c^2 \left(\frac{x_1 - x_2}{x_1 x_2}\right)}{x_2 - x_1} = -\frac{c^2 (x_1 - x_2)}{(x_1 x_2)(x_2 - x_1)} = -\frac{c^2}{x_1 x_2} \]

### Step 3: Use the point-slope form to write the equation of the chord
Using the point-slope form of the equation of a line, we have:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m\) and \(y_1\):
\[ y - \frac{c^2}{x_1} = -\frac{c^2}{x_1 x_2}(x - x_1) \]

### Step 4: Rearranging the equation
Cross-multiplying to eliminate the fraction:
\[ x_1 x_2 (y - \frac{c^2}{x_1}) = -c^2 (x - x_1) \]
Expanding both sides:
\[ x_1 x_2 y - c^2 x_2 = -c^2 x + c^2 x_1 \]
Rearranging gives:
\[ x_1 x_2 y + c^2 x = c^2 (x_1 + x_2) \]

### Step 5: Final equation
Dividing through by \(c^2\):
\[ \frac{x_1 x_2 y}{c^2} + \frac{x}{c^2} = \frac{x_1 + x_2}{c^2} \]
Thus, the equation of the chord can be written as:
\[ \frac{y}{\frac{c^2}{x_1 x_2}} + \frac{x}{c^2} = \frac{1}{\frac{1}{x_1} + \frac{1}{x_2}} \]
This simplifies to:
\[ \frac{y}{\frac{c^2}{x_1 + x_2}} + \frac{x}{c^2} = 1 \]

### Final Answer
The equation of the chord joining the points \((x_1, y_1)\) and \((x_2, y_2)\) on the rectangular hyperbola \(xy = c^2\) is:
\[ \frac{y}{\frac{c^2}{x_1 + x_2}} + \frac{x}{c^2} = 1 \]