Question

For what value k , do the equations $3x-y+8=0$ and $6x-ky+16=0$ reperesent coincident lines ?
(a) $\frac{1}{2}$ (b) $-\frac{1}{2}$ (c) 2 (d) $-2$

Answer 1

The given system of equations is
$$\begin{gathered} 3x-y+8=0 \\ 6x-ky+16=0 \end{gathered}$$
We know that the lines
$$\begin{gathered} a_{1}x+b_{1}y+c_1=0 \\ {a}_{2}x+b_{2}y+c_2=0 \end{gathered}$$
are coincident iff
$\frac{a_1}{a_2}=\frac{{\displaystyle b_1}}{{\displaystyle b_2}}=\frac{{\displaystyle c_1}}{{\displaystyle c_2}}$
$$\begin{gathered} \therefore \frac{3}{6}=\frac{{\displaystyle -1}}{{\displaystyle -k}}=\frac{8}{16} \\ \Rightarrow \frac{1}{2}=\frac{1}{k}=\frac{1}{2} \\ \Rightarrow k=2 \end{gathered}$$
Thus, the value of k = 2.

Hence, the correct answer is option (c).