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If the circle x² + y² = a² intersects the rectangular hyperbola xy = c² in four points P(x1,y1), Q(x2,y2), R(x3,y3), S(x4,y4), then which of the following is true?

1. x1 + x2 + x3 + x4 = 0
2. y1 + y2 + y3 + y4 = a
3. x1 x2 x3 x4 = a⁴
4. y1 y2 y3 y4 = a⁴

Correct answer: x1 + x2 + x3 + x4 = 0

Solution

Substituting y = c²/x into x² + y² = a² gives x⁴ - a² x² + c⁴ = 0; absence of the x³ term means the sum of the four x-roots is 0 (and similarly for y).

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