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Question
Show that the straight lines given by (2 + k) x + (1 + k) y = 5 + 7k for different values of k pass through a fixed point. Also, find that point.
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Solution
The given straight line (2 + k)x + (1 + k)y = 5 + 7k can be written in the following way:
2x + y − 5 + k (x + y − 7) = 0
This line is of the form L1 + kL2 = 0, which passes through the intersection of the lines
L1 = 0 and L2 = 0, i.e. 2x + y − 5 = 0 and x + y − 7 = 0.
Solving 2x + y − 5 = 0 and x + y − 7 = 0, we get (−2, 9), which is the fixed point.
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