In a circle of radius 5 cm, AB is a diameter and PQ is a chord that is not a diameter. The diameter AB and the chord PQ intersect at point X at right angles (AX ⟂ PQ). The segments on AB satisfy AX : BX = 3 : 2, where X lies between A and B. What is the length (in cm) of chord PQ?

Introduction:
This geometry problem involves a circle with a diameter and a chord intersecting at right angles. Knowing the ratio in which the intersection point divides the diameter, and the radius of the circle, allows us to use coordinate geometry or chord length formulas to find the length of the chord.

Given Data / Assumptions:

• Circle radius R = 5 cm.
• AB is a diameter, so AB = 2R = 10 cm.
• Chord PQ is perpendicular to AB at point X.
• X lies on AB with AX : XB = 3 : 2.
• We must find the length of chord PQ.

Concept / Approach:
We can place the circle in a coordinate system with center at the origin. Take AB as a horizontal diameter. Then we can locate point X using the given ratio along AB. Since PQ is perpendicular to AB at X, it will be a vertical line through X. Using the circle equation x² + y² = R², we can find the y coordinates of the intersection points and hence the chord length.

Step-by-Step Solution:
Let the center O be at (0, 0). Then radius R = 5, so circle equation is x² + y² = 25. Take diameter endpoints as A(−5, 0) and B(5, 0). AX : XB = 3 : 2 and AX + XB = 10 ⇒ 3k + 2k = 10 ⇒ 5k = 10 ⇒ k = 2. So AX = 6 and XB = 4. Starting from A(−5, 0), moving 6 units right gives X at (−5 + 6, 0) = (1, 0). Chord PQ is perpendicular to AB, so it lies on the vertical line x = 1. Substitute x = 1 into x² + y² = 25 ⇒ 1 + y² = 25 ⇒ y² = 24 ⇒ y = ±√24 = ±2√6. Thus P = (1, 2√6) and Q = (1, −2√6). Length of PQ is vertical distance: |2√6 − (−2√6)| = 4√6 cm.

Verification / Alternative check:
Use the formula for chord length when distance from center to chord is known. Distance OX = 1. Chord length L = 2√(R² − OX²) = 2√(25 − 1) = 2√24 = 4√6 cm. This matches the coordinate calculation.

Why Other Options Are Wrong:
Lengths 2√13 cm, 5√3 cm, and 6√5 cm correspond to different distances from the center or radius values and do not match the condition that OX = 1 and R = 5. Only 4√6 cm satisfies the geometry and the circle equation simultaneously.

Common Pitfalls:
Errors can occur when placing coordinates for A and B, or when translating the ratio AX : XB to actual segment lengths. Another error is miscalculating the distance from the center to the chord. Using either coordinate geometry or direct chord length formulas carefully avoids these issues.

Final Answer:
The length of chord PQ is 4√6 cm.