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InterviewSolution
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If 2x2 + 7xy + 3y2 + 8x + 14y + λ = 0 represents a pair of straight lines, the value of λ is1. 22. 43. 64. 8 |
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Answer» 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 |
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