Question

find the value of $k$. if the following equation represent a pair of lines $3x^2+10xy+3y^2+16y+k=0$.

Answer 1

Given equation is $3x^2+10xy+3y^2+16y+k=0$

Comparing it with $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

We get, $a=3,h=5,b=3,g=0,f=8,c=k$

The equation represents pair of straight lines.

$\therefore$ $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$\Rightarrow$ $3\times 3\times k+2\times 8\times 0\times 5-3(8{)}^2-3(0{)}^2-k(5{)}^2=0$

$\Rightarrow$ $9k+0-192-0-25k=0$

$\Rightarrow$ $-16k-192=0$

$\Rightarrow$ $-16k=192$

$\therefore$ $k=-12$