If the normal at four points `P_(i)(x_(i), (y_(i)) l, I = 1, 2, 3, 4` on the rectangular hyperbola `xy = c^(2)` meet at the point `Q(h, k),` prove that
`x_(1) + x_(2) + x_(3) + x_(4) = h, y_(1) + y_(2) + y_(3) + y_(4) = k`
`x_(1)x_(2)x_(3)x_(4) =y_(1)y_(2)y_(3)y_(4) =-c^(4)`

Explanation

To prove the given statements, we will start by deriving the equation of the normal to the rectangular hyperbola \(xy = c^2\) at the points \(P_i(x_i, y_i)\), where \(y_i = \frac{c^2}{x_i}\).

### Step 1: Equation of the Normal
The equation of the normal to the hyperbola at the point \(P_i\) can be derived as follows:

1. The coordinates of the point on the hyperbola are given by:
\[
P_i = \left(c t_i, \frac{c^2}{c t_i}\right) = \left(c t_i, \frac{c}{t_i}\right)
...