NIMCET 2016 — Mathematics PYQ

It will represent a pair of straight lines, if discriminant $(\Delta )$ of this equation equals zero $(\Delta =0)$  

**Discriminant:**  
$\Delta =\left|\begin{array}{ccc}a& amp;h& amp;g\\ h& amp;b& amp;f\\ g& amp;f& amp;c\end{array}\right|$  

$=abc+2fgh-a{f}^2-b{g}^2-c{h}^2$

\begin{aligned}
Calculation:
 & \mathrm{Given:~Second~degree~equation,~2x^{2}+7xy+3y^{2}+8x+14y+\lambda=0} \\
 & \mathrm{Compare~with~second-degree~equation~ax^{2}+by^{2}+2hxy+2gx+2fy+c=} \\
 & \mathrm{So,~a=2,~b=3,~h=\frac{7}{2},~g=4,~f=7~and~c=\lambda} \\
 & \text{Given equation represents a pair of straight lines} \\
 & \mathrm{So,\Delta=abc+2fgh-af^{2}-bg^{2}-ch^{2}=0} \\
 & \Rightarrow2\times3\times\lambda+2\times7\times4\times\frac{7}{2}-2\times(7)^{2}-3\times(4)^{2}-\lambda\times(\frac{7}{2})^{2}=0 \\
 & \Rightarrow6\lambda+196-98-48-\frac{49\lambda}{4}=0 \\
 & \Rightarrow50-\frac{25\lambda}{4}=0 \\
 & \Rightarrow200-25\lambda=0 \\
 & \therefore\lambda=8
\end{aligned}