Question

The value of k for which $2x^2+7\mathrm{xy}+3y^2+8x+14y+k=0$represents a pair of straight lines, is:

• A. 2
• B. 4
• C. 8
• D. 16

Answer 1

The correct option is C 8

${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+c=0\mathrm{represents}a\mathrm{pair}\mathrm{of}\mathrm{lines}\mathrm{if}\left(\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0\Rightarrow \left(\begin{array}{ccc}2& \frac{7}{2}& 4\\ \frac{7}{2}& 3& 7\\ 4& 7& k\end{array}\right|=0\Rightarrow 2\left(3k-49\right)-\frac{7}{2}\left(\frac{7k}{2}-28\right)+4\left(\frac{49}{2}-12\right)\Rightarrow 6k-98-\frac{49k}{4}+98+98-48=0\Rightarrow \frac{49k}{4}-6k-98+48=0\Rightarrow \frac{25k}{4}=50\Rightarrow k=8$