Question

Consider a general equation of degree 2, as $\lambda x^2 - 10xy + 12y^2 + 5x - 16y - 3 = 0$. The value of '$\lambda$' for which the line pair represents a pair of straight lines is

• A. 1
• B. 2
• C. 3/2
• D. 3

Answer 1

Answer: (B)

2

Answer 2

The general equation of a second-degree curve is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. For the given equation $\lambda x^2 - 10xy + 12y^2 + 5x - 16y - 3 = 0$, we have: $a = \lambda$, $2h = -10 \implies h = -5$, $b = 12$, $2g = 5 \implies g = 5/2$, $2f = -16 \implies f = -8$, $c = -3$.

The condition for the general second-degree equation to represent a pair of straight lines is that the determinant of the coefficients is zero: $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0$ Substituting the coefficients: $\begin{vmatrix} \lambda & -5 & 5/2 \\ -5 & 12 & -8 \\ 5/2 & -8 & -3 \end{vmatrix} = 0$ Expanding the determinant: $\lambda[(12)(-3) - (-8)(-8)] - (-5)[(-5)(-3) - (-8)(5/2)] + (5/2)[(-5)(-8) - (12)(5/2)] = 0$ $\lambda[-36 - 64] + 5[15 - (-20)] + (5/2)[40 - 30] = 0$ $\lambda[-100] + 5[35] + (5/2)[10] = 0$ $-100\lambda + 175 + 25 = 0$ $-100\lambda + 200 = 0$ $100\lambda = 200$ $\lambda = 2$