Which among the following equations represents a pair of straight lines?
3x2 - 4xy - 7y2 = 0
Equations of the form ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 can be the equation of a conic (Circle, Parabola, Ellipse or Hyperbola) or pair of straight lines.
To decide if it represents a conic or pair of straight lines, we will find
△ = abc + 2fgh − af2 − bg2 − ch2 = 0
If △ = 0, the given equation will represent a pair of straight lines. We will find △ for each equation and decide.
1) x2 − y2 + x − y + 1 = 0
a = 1,b = −1,h = 0,g = 12,f = −12,c = 1
⇒ △ = 1 × 1 × 1 + 0 − (−12)2 + (12)2 −1 × 0 = 1
△ ≠ 0⇒ conic
2) x2 − 2x − y + 3 = 0
a=1,b=0,h=0,g=−1,f=−12,c=3
△ = 0 + 0 − 1 × (−12)2 − 0 − 0
=−14
△ ≠ 0⇒ conic
3) x2 + 3xy + 2y2 − x − 4y − 6 = 0
a = 1,b=2,h=32,g=−12,f=−2,c=−6
△ = 1 × 2 × 32 + 2 × −2 × −12 × 32 −1 × (−2)2 −2 × (−12)2 − (−6) × (32)2
3+3−4−12 + 272
=9−4+13
△ ≠ 0⇒ conic
4) 3x2 − 4xy − 7y2 = 0
a=3,b=−7,h=−2,g=0,f=c=0
△ = 0 ⇒ pair of straight lines
So, only the last equation represents a pair of straight lines.