Conditions for pair of straight lines
If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent a pair of lines, then
i) abc + 2fgh – af 2 – bg2 – ch2 = 0 and
ii) h2 ≥ ab;   g2 ≥ ac;   f 2 ≥ bc.

Proof:

Let ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents the lines
l1x + m1y + n1 = 0 ----- (1),
l2x + m2y + n2 = 0 ----- (2)
∴ (l1x + m1y + n1) (l2x + m2y + n2) = ax2 + 2hxy + by2 + 2gx + 2fy + c
Comparing the co-efficient on both sides, we get l1l2 = a, m1m2 = b, n1n2 = c
l1m2 + l2m1 = 2h, l1n2 + l2n1 = 2g, m1n2 + m2n1 = 2f
i) (2f) (2g) (2h) = (m1n2 + m2n1) (l1n2 + l2n1) (l1m2 + l2m1)
= (m1n2 + m2n1) (l12m2n2 + 11l2m1n2 + 11l2m2n1 + l22m1 n1)
= l12m1m2n22 + 11l2m12n22 + 11l2m1m2n1n2 + 122m12n1n2 + 112m22n1n2 + 11l2m1m2n1n2 + 11l2m22n12 + l22m1m2n12
= l1l2 (m12n22 + m22n12) + m1m2 (l12n22 + l22n12) + n1n2 (l12m22 + l22m12) + 2 l1l2 m1m2 n1n2
= l1l2 [(m1n2 + m2n1)2 – 2 m1n2m2n1 ] + m1m2 [(l1n2 + l2n1)2 – 2 l1n2l2n1 ] + n1n2 [(l1m2 + l2m1)2 – 2 l1m2l2m1 ] + 2 l1l2 m1m2 n1n2
= a [(2f)2 – 2bc] + b [(2g)2 – 2ac] + c [(2h)2 – 2ab] + 2abc
= a [4f 2 – 2bc] + b [4g2 – 2ac] + c [4h2 – 2ab] + 2abc
= 4af 2 – 2abc + 4bg2 – 2abc + 4ch2 – 2abc + 2abc
= – 4abc + 4af 2 + 4bg2 + 4ch2
⇒ 8fgh = – 4abc + 4af 2 + 4bg2 + 4ch2
⇒ 8fgh + 4abc – 4af 2 – 4bg2 – 4ch2 = 0
⇒ abc + 2fgh – af 2 – bg2 – ch2 = 0
ii) 4(h2 – ab) = (2h)2 – 4ab
= (l1m2 + l2m1)2 – 4 l1l2m1m2
= (l1m2 – l2m1)2 ≥ 0
∴ 4(h2 – ab) ≥ 0
⇒ h2 – ab ≥ 0
⇒ h2 ≥ ab
Similarly we can prove g2 ≥ ac and f 2 ≥ bc.