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For which value(s) of p do the lines p(p²+1)x - y + q = 0 and (p² + 1)x + (p² + 1)y + 2q = 0 both stand perpendicular to the same line?

1. exactly one value of p
2. exactly two values of p
3. more than two values of p
4. no value of p

Correct answer: exactly one value of p

Solution

The lines are perpendicular to the same line if their slopes are negative reciprocals of each other. This condition leads to a unique relationship between the coefficients of the lines, resulting in exactly one value of p that satisfies this requirement.

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