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Consider the two curves$$C_1 : y^2 = 4x$$$$C_2 : x^2 + y^2 - 6x + 1 = 0$$Then,
$$C_1$$ and $$C_2$$ touch each other only at one point
$$C_1$$ and $$C_2$$ touch each other exactly at two points
$$C_1$$ and $$C_2$$ intersect (but do not touch) at exactly two points
$$C_1$$ and $$C_2$$ neither intersect nor touch each other
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