Question

The second-degree general equation S ≡ ax^{2 +} 2hxy + by² + 2gx + 2fy + c = 0 represents a pair of lines if and only if 

1.   abc + 2fgh − af ² − bg² − ch² = 0. 

2.   h² ≥ rab, g²  ≥ ac and f² ≥ bc.

Answer 1

Suppose S = 0 represents pair of lines and let the lines be l₁x + m₁y + n₁ = 0 and l₂x + m₂y + n₂ = 0. Therefore

 S ≡ (l₁x + m₁y + n₁) (l₂x + m₂y + n₂)

 Equating the corresponding coefficients on both sides, we have l ₁l₂ = a, l₁m₂ + l₂m₁ = 2h and m₁m₂ = b, l₁n₂ + l₂n₁ = 2g, m₁n₂ + m₂n₁ =  2f, n₁n₂ = c.

Therefore 

2fgh =  af² + bg² + ch² - abc or  abc + 2 fgh -  af² - bg² -  ch² 

Generally, the number of abc + 2 fgh - af² -  bg² - ch² is denoted by Δ. Therefore, Δ = 0. Also

Similarly, g² ≥ ca and f² ≥ bc. 

The proof of the converse part is a bit lengthy and beyond the scope of this book. Hence, we assume the validity of the converse part.