`P N` is the ordinate of any point `P` on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` and `AA '` is its transvers axis. If `Q` divides `A P` in the ratio `a^2: b^2,` then prove that `N Q` is perpendicular to `A^(prime)Pdot`

Explanation

To solve the problem, we need to prove that the line segment \( NQ \) is perpendicular to the line segment \( A'P \) given the conditions stated in the problem. Let's break down the solution step by step.

### Step 1: Understand the Hyperbola and Points
The equation of the hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Let \( P \) be a point on the hyperbola with coordinates \( P(a \sec \theta, b \tan \theta) \). The points \( A \) and \( A' \) are given as \( A(a, 0) \) and \( A'(-a, 0) \), respectively. The point \( N \) is defined as \( N(x, 0) \) where \( x \) is the x-coordinate of point \( P \).
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