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Question

Prove that the family of lines represented by x(1+λ)+y(2λ)+5=0,λ being arbitary, pass through a fixed point .Also,find the fixed point.

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Solution

x(1+λ)+y(2λ)+5=0

x+xλ+2yλy+5=0

λ(xy)+(x+2y+5)=0

(x+2y+5)+λ(xy)=0

This is of the form L1+λL2=0

So it represents a line passing through the intersenction of xy=0 and x+2y=5

Solving the two equations,

we get (53,53) which is the fixed point through which the given family of lines passes for any value of λ.


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