Q89 / 160 · AP EAMCET 2008 · Mathematics

The value of $\lambda$ such that $\lambda x^2-10 x y+12 y^2+5 x-16 y-3=0$ represents a pair of straight lines, is

  • A
    $1$
  • B
    $-1$
  • Correct answer
    C
    $2$
  • D
    $-2$

Solution

Given pair of lines is $$ \lambda x^2-10 x y+12 y^2+5 x-16 y-3=0 $$ On comparing with $$ a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 $$ we get, $$ a=\lambda, h=-5, b=12, g=\frac{5}{2}, f=-8, c=-3 $$ Condition for represents a pair of lines is $$ \begin{aligned} & a b c+2 f g h-a f^2-b g^2-c h^2=0 \\ & \lambda \times 12 \times(-3)+2(-8)\left(\frac{5}{2}\right)(-5)-\lambda(-8)^2 \\ & -12\left(\frac{5}{2}\right)^2+3(-5)^2=0 \\ & \Rightarrow-36 \lambda+200-64 \lambda-75+75=0 \\ & \Rightarrow \quad 100 \lambda=200 \Rightarrow \lambda=2 \\ & \end{aligned} $$

Hence, the correct answer is option: 'C'.

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