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Question
Prove that the family of lines represented by x (1 + λ) + y (2 − λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.
Solution
The given family of lines can be written as
x + 2y + 5 + λ (x − y) = 0
This line is of the form L1 + λL2 = 0, which passes through the intersection of L1 = 0 and L2 = 0.
⇒ x + 2y + 5 = 0
⇒ x − y = 0
Now, solving the lines: \[\left( - \frac{5}{3}, - \frac{5}{3} \right)\] This is a fixed point.
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