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The two circles x² + y² + 2ax + c² = 0 and x² + y² + 2by + c² = 0 touch each other. Which relation between a, b and c must hold?

1. 1/a² + 1/b² = 1/c²
2. 1/a² + 1/b² = 1/(2c²)
3. a² + b² = c²
4. 1/a² - 1/b² = 1/c²

Correct answer: 1/a² + 1/b² = 1/c²

Solution

Centres are (-a,0) and (0,-b) with radii sqrt(a² - c²) and sqrt(b² - c²). The touching condition d² = (r1 +- r2)² simplifies to 1/a² + 1/b² = 1/c².

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