Two circles touch each other externally. The sum of their areas is 130π square centimetres and the distance between their centres is 14 cm. What is the radius of the larger circle?

Introduction / Context:
This question checks understanding of circle geometry, including area of a circle and basic algebra with two unknown radii. It also uses the condition that two circles touch externally, which gives a relationship between the radii and the distance between the centres.

Given Data / Assumptions:
- Two circles touch externally, so the distance between their centres equals the sum of their radii.
- Let the radii be r1 and r2, where r1 is the radius of the larger circle and r2 is the smaller one.
- The sum of their areas is 130π square centimetres.
- The distance between the centres is 14 cm, so r1 + r2 = 14.
- We need to find r1, the radius of the bigger circle.

Concept / Approach:
The area of a circle with radius r is π * r^2. The total area of two circles is therefore π * r1^2 + π * r2^2, which is given to be 130π. This gives r1^2 + r2^2 = 130. Combined with the touching condition r1 + r2 = 14, we have a system of two equations in r1 and r2 that can be solved using algebraic identities.

Step-by-Step Solution:
Step 1: From the area condition, write π * r1^2 + π * r2^2 = 130π. Step 2: Divide both sides by π to get r1^2 + r2^2 = 130. Step 3: From the external touching condition, write r1 + r2 = 14. Step 4: Square the second equation: (r1 + r2)^2 = 14^2 = 196. Step 5: Expand the square: r1^2 + 2 * r1 * r2 + r2^2 = 196. Step 6: Substitute r1^2 + r2^2 = 130 into this: 130 + 2 * r1 * r2 = 196. Step 7: Simplify to obtain 2 * r1 * r2 = 196 − 130 = 66, so r1 * r2 = 33. Step 8: Now r1 and r2 satisfy r1 + r2 = 14 and r1 * r2 = 33, so they are roots of the quadratic x^2 − 14x + 33 = 0. Step 9: Factor this quadratic: x^2 − 14x + 33 = (x − 11)(x − 3). Step 10: The roots are x = 11 and x = 3. So the radii are 11 cm and 3 cm. Step 11: The larger radius is therefore 11 cm.

Verification / Alternative Check:
Check the area condition: r1^2 + r2^2 = 11^2 + 3^2 = 121 + 9 = 130, which matches the given total area divided by π. Check the distance condition: r1 + r2 = 11 + 3 = 14 cm, which matches the distance between centres. Both conditions are satisfied, confirming the solution.

Why Other Options Are Wrong:
Option a, 22 cm, is too large and would make r1 + r2 exceed 14 cm for any positive r2, violating the touching condition.
Option c, 33 cm, is even larger and clearly impossible when the distance between centres is only 14 cm.
Option d, 44 cm, is far beyond the possible value and does not satisfy either given condition.

Common Pitfalls:
Some students attempt to guess radii directly from the area sum without using the condition that the circles touch externally. Others may forget to divide by π when equating areas, leading to more complicated expressions. Using the identity for (r1 + r2)^2 simplifies the algebra significantly and is a useful technique whenever sums and squares of variables are involved.

Final Answer:
The radius of the larger circle is 11 cm.