If the equation $3 x^{2}+10 x y+3 y^{2}+16 y+k=0$ represents a pair of lines, then the value of $k$ is
If the equation $3 x^{2}+10 x y+3 y^{2}+16 y+k=0$ represents a pair of lines, then
the value of $k$ is
$-21$
21
12
$-12$
Solution
Comparing the given equation with $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$, we get, $a=3, h=5, b=3, g=0, f=8, c=k$.
Now, given equation represents a pair of lines.
$\therefore a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$
$\therefore$ (3) (3) $(\mathrm{k})+2(8)(0)(5)-3(8)^{2}-3(0)^{2}-\mathrm{k}(5)^{2}=0$
$\therefore 9 \mathrm{k}+0-192-0-25 \mathrm{k}=0 \Rightarrow 16 \mathrm{k}=-192 \Rightarrow \mathrm{k}=-12$