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If the points (a³/(a − 3), (a² − 3)/(a − 1)), (b³/(b − 3), (b² − 3)/(b − 1)), and (c³/(c − 3), (c² − 3)/(c − 1)), where a, b, and c are not equal to 1, are situated on the line l₁x + m₁y + n₁ = 0, then which of the following is true?

1. The sum a + b + c equals −m/l.
2. The product ab + bc + ca equals n/l.
3. The product abc equals (m + n)/l.
4. The expression abc − (bc + ca + ab) + 3(a + b + c) equals 0.

Correct answer: The expression abc − (bc + ca + ab) + 3(a + b + c) equals 0.

Solution

The points given are situated on a line, and by using the equation of the line and the coordinates of the points, it can be derived that a specific expression involving the coefficients a, b, and c equals zero, which is a necessary condition for the points to lie on the line.

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