If `A ,B ,a n dC`
are three points on the hyperbola `x y=c^2`
such that `A B`
subtends a right angle at `C ,`
then prove that `A B`
is parallel to the normal to the hyperbola at point `Cdot`
Text Solution
AI Generated Solution
To prove that the line segment \( AB \) is parallel to the normal to the hyperbola at point \( C \), we will follow these steps:
### Step 1: Define the Points on the Hyperbola
Let the points \( A \), \( B \), and \( C \) on the hyperbola \( xy = c^2 \) be represented in terms of parameters \( t_1 \), \( t_2 \), and \( t_3 \):
- Point \( A \) has coordinates \( (ct_1, \frac{c^2}{t_1}) \)
- Point \( B \) has coordinates \( (ct_2, \frac{c^2}{t_2}) \)
- Point \( C \) has coordinates \( (ct_3, \frac{c^2}{t_3}) \)
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