Correct Answer – Option 4 : 8

Concept:

Let second-degree equation be ax2 + by2 + 2hxy + 2gx + 2fy + c = 0

It will represent a pair of straight lines, If discriminant (Δ) of this equation equal to zero (Δ = 0) 

Discriminant = Δ = \(\left| {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{h}}&{\rm{g}}\\ {\rm{h}}&{\rm{b}}&{\rm{f}}\\ {\rm{g}}&{\rm{f}}&{\rm{c}} \end{array}} \right| = {\rm{abc}} + 2{\rm{fgh}} – {\rm{a}}{{\rm{f}}^2} – {\rm{b}}{{\rm{g}}^2} – {\rm{c}}{{\rm{h}}^2}\)

Calculation:

Given: Second degree equation, 2x2 + 7xy + 3y2 + 8x + 14y + λ = 0

Compare with second-degree equation ax2 + by2 + 2hxy + 2gx + 2fy + c = 0

So, a = 2, b = 3, h = \(\frac 72\), g = 4, f = 7 and c = λ

Given equation represents a pair of straight lines

So, Δ = abc + 2fgh – af2 – bg2 – ch² = 0

⇒ 2 × 3 × λ + 2 × 7 × 4 × \(\frac 72\) – 2 × (7)² – 3 × (4)2 – λ × (\(\frac 72\))2 = 0

⇒ 6λ + 196 – 98 – 48 – \(\frac {49λ}{4}\) = 0

⇒ 50 – \(\frac {25λ}{4}\)= 0

⇒ 200 – 25λ = 0

∴ λ = 8