Prove that product of parameters of four concyclic points on the hyperbola `xy=c^(2)` is 1. Also, prove that the mean of these four concyclic points bisects the distance between the centres of the hyperbola and the circle.
Text Solution
AI Generated Solution
To prove that the product of the parameters of four concyclic points on the hyperbola \(xy = c^2\) is 1, and that the mean of these points bisects the distance between the centers of the hyperbola and the circle, we can follow these steps:
### Step 1: Define the Hyperbola and Points
The equation of the hyperbola is given by:
\[
xy = c^2
\]
Let the four concyclic points on this hyperbola be denoted as \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), and \((x_4, y_4)\). From the hyperbola's equation, we can express the \(y\)-coordinates in terms of the \(x\)-coordinates:
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