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Question
Find the condition that the equation ay2 + bxy + ex + dy = 0 may represent a pair of lines.
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Solution
Comparing the equation ay2 + bxy + ex + dy = 0 with Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0, we get,
A = 0, H = `"b"/2`, B = a, G = `"e"/2`, F = `"d"/2`, C = 0
The given equation represents a pair of lines,
if `|("A","H","G"),("H","B","F"),("G","F","C")| = 0`
i.e. if `|(0,"b"/2,"e"/2),("b"/2,"a","d"/2),("e"/2,"d"/2, 0)| = 0`
i.e. if `0 - "b"/2(0 - "ed"/4) + "e"/2("bd"/4 - "ae"/2) = 0`
i.e. if `"bed"/8 + "bed"/8 - "ae"^2/4 = 0`
i.e. if bed - ae2 = 0
i.e. if e(bd - ae) = 0
i.e. if e = 0 or bd - ae = 0
i.e. if e = 0 or bd = ae
This is the required condition.
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