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Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0 - Mathematics and Statistics

Sum

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0 

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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`

∴ `(16"b")/25 - "8h"/5 + "a" = 0`

∴ 16b − 40h + 25a = 0

∴ 25a + 16b − 40h = 0

This is the required condition.

Concept: Homogeneous Equation of Degree Two
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