Question

Find the condition that the equation ay² + bxy + ex + dy = 0 may represent a pair of lines. 

Answer 1

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.