Question

Find $p$ for which given equation has unique solution
$4x+py+8=0$
$2x+2y+2=0$

Answer 1

$4x+py+8=0$ ------- ( 1 )

$2x+2y+2=0$ ------- ( 2 )

$\Rightarrow$ $4x+py+8=0$ is comparing with $a_{1}x+b_{1}y+c_1=0$

$\therefore$ We get, $a_1=4,b=p,c_1=8$

$\Rightarrow$ $2x+2y+2=0$ is comparing with $a_{2}x+b_{2}y+c_2=0$

$\therefore$ We get, $a_2=2,b_2=2,c_2=2$

$\Rightarrow$ So, $a_1=4,b_1=p,c_1=8$ and $a_2,b_2=2,c_2=2$

It is given that equation has unique number of solutions,

$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

$\Rightarrow$ $\frac{4}{2}\ne \frac{p}{2}$

$\Rightarrow$ $p\ne \frac{4}{2}\times 2$

$\Rightarrow$ $p\ne 4$

$\Rightarrow$ For all values of $p$, except $4$, the given pair of equations will have a unique solution.