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Question:

If the three distinct lines x+2ay+a=0, x+3by+b=0 and x+4ay+a=0 are concurrent, then the point (a,b) lies on a: Circle, Straight line, Hyperbola, Parabola?

Hyperbola

Straight line

Parabola

Circle

Solution:

|1 2a a|
|1 3b b|
|1 4a a|=0 ⇒ (3ab−ab)−a(4a−b)=0 ⇒ 2ab−a(4a−b)=0 ⇒ 2a²−ab=0 ⇒ a(2a−b)=0
Hence, it lies on a straight line.