Question : Two circles with radii \(p, q\) \((p > q)\) in cm touch externally, where \(p, q\) are natural numbers and each greater than 1. What is the value of \((p - q)\) ?
Statement-I : The sum of their areas is \(130\pi \text{ cm}^2\)
Statement-II : The distance between the centres of the two circles is 14 cm
Which one of the following is correct in respect of the above Question and the Statements ?
Analyzing Statement Sufficiency for Circle Radii Problem
The core task is to find the value of \((p - q)\) for two externally touching circles with radii \(p\) and \(q\). Key conditions are \(p > q\), and both \(p, q\) are natural numbers greater than 1.
Statement I Analysis: Sum of Areas
Statement I gives the sum of the areas:
\(\pi p^2 + \pi q^2 = 130\pi \text{ cm}^2\)
Dividing by \(\pi\), we get:
\(p^2 + q^2 = 130\)
We look for natural number pairs \((p, q)\) where \(p > q > 1\) that satisfy this equation.
- Possible pairs are \((11, 3)\) because \(11^2 + 3^2 = 121 + 9 = 130\). Here, \(p - q = 11 - 3 = 8\).
- Another possible pair is \((9, 7)\) because \(9^2 + 7^2 = 81 + 49 = 130\). Here, \(p - q = 9 - 7 = 2\).
Since Statement I alone yields two different values for \((p - q)\), it is insufficient to answer the question.
Statement II Analysis: Distance Between Centers
Statement II states the distance between the centers is 14 cm. For circles touching externally, this distance is the sum of their radii:
\(p + q = 14\)
We list pairs \((p, q)\) satisfying \(p > q > 1\) and \(p + q = 14\).
- \((12, 2) \implies p - q = 10\)
- \((11, 3) \implies p - q = 8\)
- \((10, 4) \implies p - q = 6\)
- \((9, 5) \implies p - q = 4\)
- \((8, 6) \implies p - q = 2\)
Statement II alone results in multiple possible values for \((p - q)\), hence it is insufficient.
Combined Analysis of Statements I and II
To answer the question, both statements must hold true simultaneously.
- Condition from Statement I: \(p^2 + q^2 = 130\). Possible pairs \((p, q)\) are \((11, 3)\) and \((9, 7)\).
- Condition from Statement II: \(p + q = 14\).
We check the pairs from Statement I against Statement II:
- Pair \((11, 3)\): \(11 + 3 = 14\). This satisfies Statement II. The difference \(p - q = 11 - 3 = 8\).
- Pair \((9, 7)\): \(9 + 7 = 16\). This does not satisfy Statement II (\(p + q = 14\)).
Only the pair \((11, 3)\) satisfies both conditions. This provides a unique value for \(p - q\), which is 8.
Therefore, the question can be answered only when both statements are used together.
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