P and Q are two points on the rectangular hyperbola `xy = C^(2)` such that the abscissa of P and Q are the roots of the equations `x^(2) - 6x - 16 = 0`. Equation of the chord joining P and Q is

Options

A. `16x-c^(2)y=6c^(2)`
B. `c^(2)x-16y=c^(2)`
C. `c^(2)x-16y=6c^(2)`
D. `c^(2)x-16y=6c^(2)`

Answer

To find the equation of the chord joining points P and Q on the rectangular hyperbola \(xy = C^2\), where the abscissas of P and Q are the roots of the equation \(x^2 - 6x - 16 = 0\), we will follow these steps:

### Step 1: Find the roots of the quadratic equation
The given quadratic equation is:
\[ x^2 - 6x - 16 = 0 \]
We can find the roots using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -6\), and \(c = -16\).

### Step 2: Calculate the discriminant
First, we calculate the discriminant:
\[ D = b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100 \]

### Step 3: Find the roots
Now, substituting back into the quadratic formula:
\[ x = \frac{6 \pm \sqrt{100}}{2} = \frac{6 \pm 10}{2} \]
This gives us two roots:
\[ x_1 = \frac{16}{2} = 8, \quad x_2 = \frac{-4}{2} = -2 \]

### Step 4: Identify points P and Q
The points P and Q on the hyperbola can be represented in parametric form as:
\[ P(t_1) = (C t_1, \frac{C^2}{t_1}), \quad Q(t_2) = (C t_2, \frac{C^2}{t_2}) \]
Here, \(t_1\) and \(t_2\) correspond to the roots we found, which are 8 and -2.

### Step 5: Calculate \(t_1 t_2\)
Now, we calculate:
\[ t_1 t_2 = 8 \cdot (-2) = -16 \]

### Step 6: Write the equation of the chord
The equation of the chord joining points \(P\) and \(Q\) on the hyperbola \(xy = C^2\) is given by:
\[ y - y_1 = m(x - x_1) \]
Where \(m\) is the slope of the chord. However, we can also express it in the form:
\[ x + y = t_1 t_2 + C^2 \]
Substituting \(t_1 t_2 = -16\):
\[ x + y = -16 + C^2 \]

### Final Result
Thus, the equation of the chord joining points P and Q is:
\[ x + y = C^2 - 16 \]