# 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.

Given:

Lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary

To prove:

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.

Explanation:

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:
This is a fixed point.

Hence proved.

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