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Question

If the equation 6x2αxy3y224x+3y+β=0 represents a pair of straight lines that intersect on the xaxis, then the value of 20αβ is

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Solution

As the lines intersect at the xaxis, let point of intersection of lines be P(k,0)
Therefore point P must satisfy the given equation
6k224k+β=0

There is only one such k possible for the intersection of the two lines and so the above equation must have repreated roots.
b24ac=0
β=24 and k=2

Substitute (2,y) back into the equation
6(2)2α(2)y3y224(2)+3y+24=0y2+(32α)y=0
But we know that y=0 is the only solution to the above equation, so even this equation has repeated roots.
b24ac=0α=32

Hence 20αβ=20×3224=6

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