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Question
If the line 4x - 5y = 0 coincides with one of the lines given by ax2 + 2hxy + by2 = 0, then show that 25a + 40h + 16b = 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 = 4/5`
∴ m = `4/5` is a root of the auxiliary equation bm2 + 2hm + a = 0.
∴ `"b"(4/5)^2 + 2"h"(4/5) + "a" = 0`
∴ `"16b"/25 + "8h"/5 + "a" = 0`
∴ 16b + 40h + 25a = 0 i.e.
∴ 25a + 40h + 16b = 0
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