Let's analyze the problem step by step.

Given:

• A circle with radius $r$.
• Two perpendicular chords intersect at point $P$ inside the circle.
• The lengths of the chords are $a$ and $b$.

Let's denote the line segments created by the intersection of the chords as follows:

• Let the chord of length $a$ be divided into two segments $x$ and $a-x$.
• Let the chord of length $b$ be divided into two segments $y$ and $b-y$.

The goal is to find the product of the lengths of these segments: $x \cdot (a-x) \cdot y \cdot (b-y)$.

Step 1: Use the Power of a Point Theorem

By the Power of a Point Theorem (or Intersecting Chords Theorem), for any point $P$ inside the circle, the products of the lengths of the segments of each chord are equal. That is: $x \cdot (a - x) = y \cdot (b - y)$

Step 2: Calculate the Required Product

We need to find $x \cdot (a-x) \cdot y \cdot (b-y)$.

Let $x \cdot (a-x) = p$ and $y \cdot (b-y) = q$.

Since $p = q$, the product is: $x \cdot (a-x) \cdot y \cdot (b-y) = p \cdot q = p^2$