The value of $\lambda$ with $|\lambda|

Question

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

Aicharya · Accepted Answer

Given equation is $2 x^2-10 x y+12 y^2+5 x+\lambda y-3=0$ Here, $a=2, h=-5, b=12, g=\frac{5}{2}, f=\frac{\lambda}{2}, c=-3$ For pair of lines $\left|\begin{array}{lll}a & h & g \ h & b & f \ g & f & c\end{array}ight|=0$ $\begin{array}{cc}\Rightarrow & \left|\begin{array}{rrr}2 & -5 & 5 / 2 \ -5 & 12 & \lambda / 2 \ 5 / 2 & \lambda / 2 & -3\end{array}ight|=0 \ \Rightarrow & 2\left(-36-\frac{\lambda^2}{4}ight)+5\left(15-\frac{5 \lambda}{4}ight) \ & +\frac{5}{2}\left(\frac{-5 \lambda}{2}-30ight)=0 \ \Rightarrow & -72-\frac{\lambda^2}{2}+75-\frac{25 \lambda}{4}-\frac{25 \lambda}{4}-75=0 \ \Rightarrow & \lambda^2+25 \lambda+144=0 \ \Rightarrow & (\lambda+9)(\lambda+16)=0 \ \Rightarrow & \lambda=-9 \quad(\because|\lambda| < 16)\end{array}$

The value of $\lambda$ with $|\lambda| < 16$ such that $2 x^2-10 x y+12 y^2+5 x+\lambda y-3=0$ represents a…

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