For how many values of the real parameter a do the pair of lines represented by 3ax² + 5xy + (a² - 2)y² = 0 become perpendicular to each other?

two values of a
for all values of a
for one value of a
for no values of a

Correct answer: two values of a

Solution

The perpendicularity condition for ax² + 2hxy + by² = 0 requires a + b = 0, giving 3a + a² - 2 = 0, which has discriminant 9 + 8 = 17 > 0, yielding two distinct real values of a.

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