Question

If $u=a_{1}x+b_{1}y+c_1=0$ and $v=a_{2}x+b_{2}y+c_2=0$ and $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, then $u+kv=0$ represents

• A. $u=0$
• B. a family of concurrent lines
• C. a family of parallel lines
• D. None of these

Answer 1

The correct option is A $u=0$

Given:$u=a_{1}x+b_{1}y+c_1=0$

and $v=a_{2}x+b_{2}y+c_2=0$

We have $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\Rightarrow u=0$ and $v=0$ are the same equations

Thus, $u+kv=0$ represent $(1+k)u=0$ or $(1+k)v=0$

i.e., $u=0$ or $v=0$

Hence option$\left(a\right)$ is correct.