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Question
Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0
Solution
The auxiliary equation of the lines represented by ax2 + 2hxy + by2 = 0 is bm2 + 2hm + a = 0
Given that 4x + 5y = 0 is one of the lines represented by ax2 + 2hxy + by2 = 0
The slope of the line 4x + 5y = 0 is `- 4/5`
∴ m = `- 4/5` is a root of the auxiliary equation bm2 + 2hm + a = 0
∴ `b(- 4/5)^2 + 2h(- 4/5) + a` = 0
∴ `(16b)/25 - (8h)/5 + a` = 0
∴ 16b − 40h + 25a = 0
∴ 25a + 16b – 40h = 0
This is the required condition.
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