Question

If $u=a_{1}x+b_{1}y+c_1=0$ and $v=a_{2}x+b_{2}y+c_2=0$ and $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ prove that the line curve $u+kv=0$ is nothing but any of the given straight lines $u=0$ or $v=0$

Answer 1

The equations of the given lines are :
$u=a_{1}x+b_{1}y+c_1=0,v=a_{2}x+b_{2}y+c_2=0$
Also it is given that
$a_1/a_2=b_1/b_2=c_1/c_2=\lambda$, say
By using relation in $\left(1\right)$, the family of lines $u+kv=0$ becomes $\lambda v+kv=0$
or $(\lambda +k)v=0$ or $v=0$ or $u+\frac{k}{\lambda }u=0$
or $(1+\frac{k}{\lambda })u=0$ or $u=0$.