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Find the condition that the line 3x + y = 0 may be perpendicular to one of the lines given by ax2 + 2hxy + by2 = 0 - Mathematics and Statistics

Sum

Find the condition that the line 3x + y = 0 may be perpendicular to 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 

Since one line is perpendicular to the line 3x + y = 0 whose slope is `- 3/1 = - 3`

∴ slope of that line = m = `1/3`

∴ m = `1/3` is the root of the auxiliary equation  bm2 + 2hm + a = 0.

∴ `"b"(1/3)^2 + "2h"(1/3) + "a" = 0`

∴ `"b"/9 + "2h"/3 + "a" = 0`

∴ b + 6h + 9a = 0

∴ 9a + b + 6h = 0

This is the required condition.

Concept: Homogeneous Equation of Degree Two
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