Show that the equation of the chord joining two points `(x_(1),y_(1)) and (x_(2),y_(2))` on the rectangular hyperbola `xy=c^(2) is (x)/(x_(1)+x_(2)) +(y)/(y_(1)+y_(2))=1`

Answer

To show that the equation of the chord joining two points \((x_1, y_1)\) and \((x_2, y_2)\) on the rectangular hyperbola \(xy = c^2\) is given by

\[ \frac{x}{x_1 + x_2} + \frac{y}{y_1 + y_2} = 1, \]

we will follow these steps:

### Step 1: Identify the points on the hyperbola
The points \((x_1, y_1)\) and \((x_2, y_2)\) lie on the hyperbola defined by the equation \(xy = c^2\). Therefore, we can express \(y_1\) and \(y_2\) in terms of \(x_1\) and \(x_2\):
\[ y_1 = \frac{c^2}{x_1} \quad \text{and} \quad y_2 = \frac{c^2}{x_2}. \]

### Step 2: Find the slope of the chord
The slope \(m\) of the chord joining the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1}. \]
Substituting \(y_1\) and \(y_2\):
\[ m = \frac{\frac{c^2}{x_2} - \frac{c^2}{x_1}}{x_2 - x_1}. \]
Factoring out \(c^2\):
\[ m = \frac{c^2 \left(\frac{1}{x_2} - \frac{1}{x_1}\right)}{x_2 - x_1}. \]
This simplifies to:
\[ m = \frac{c^2 \left(\frac{x_1 - x_2}{x_1 x_2}\right)}{x_2 - x_1} = -\frac{c^2}{x_1 x_2}. \]

### Step 3: Use the point-slope form of the line
Using the point-slope form of the line equation, the equation of the chord can be written as:
\[ y - y_1 = m(x - x_1). \]
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
Multiplying through by \(x_1 x_2\) to eliminate the fraction:
\[ x_1 x_2 y - c^2 x_2 = -c^2 x + c^2 x_1. \]
Rearranging gives:
\[ x_1 x_2 y + c^2 x = c^2 (x_1 + x_2). \]

### Step 5: Divide by \(c^2\)
Dividing the entire equation by \(c^2\) yields:
\[ \frac{x_1 x_2 y}{c^2} + x = x_1 + x_2. \]

### Step 6: Substitute \(y_1\) and \(y_2\)
Since \(y_1 = \frac{c^2}{x_1}\) and \(y_2 = \frac{c^2}{x_2}\), we can rewrite the equation:
\[ \frac{x}{x_1 + x_2} + \frac{y}{y_1 + y_2} = 1. \]

Thus, we have shown that the equation of the chord joining the two points on the hyperbola is:
\[ \frac{x}{x_1 + x_2} + \frac{y}{y_1 + y_2} = 1. \]