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Consider the curves y = A*x² (A a positive constant) and y² + 3 = x² + 4y, with x, y real. How many points of intersection do the two curves have?

1. exactly 4
2. exactly 2
3. at least 2 but the number varies with different positive values of A
4. zero for at least one positive value of A

Correct answer: exactly 4

Solution

Substituting the upward parabola into the hyperbola gives a quadratic in y with two positive roots for every A > 0; each positive root yields x = +/- sqrt(y/A), so there are always exactly 4 intersection points.

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