Which among the following represents the combined equation of a pair of lines through point $(1,0)$ and…
Which among the following represents the combined equation of a pair of lines through point $(1,0)$ and parallel to the lines represented by $2 x^2-x y-y^2=0$.
$2 x^2-x y-2 y^2+4 x-y=6$
$2 x^2-x y-y^2-4 x+y+2=0$
$2 x^2-x y-2 y^2-4 x+y+2=0$
$2 x^2-x y-y^2-4 x-y=2$
Solution
Given line,
$2 x^2-x y-y^2=0...(i)$
$\begin{aligned} \Rightarrow \quad 2 x^2-2 x y+x y-y^2 & =0 \\ 2 x(x-y)+y(x-y) & =0 \\ (2 x+y)(x-y) & =0\end{aligned}$
Since, we have given that combined equation of the pair is parallel to $2 x^2-x y-y^2=0$
So, combined equation will be
$\left(2 x+y+k_1\right)\left(x-y+k_2\right)=0$ where $k_1$ and $k_2 \ldots(ii)$
are constants.
This equation satisfies (1, 0).
$\begin{array}{rrrl}\therefore & \left(2 \times 1+0+k_1\right)\left(1-0+k_2\right) & =0 \\ \Rightarrow & k_2=-1 \text { and } k_1 & =-2\end{array}$
On putting the values of $k_1$ and $k_2$ in Eq. (ii)
$
\begin{aligned}
(2 x+y-2)(x-y-1) & =0 \\
\Rightarrow \quad 2 x^2-x y-y^2-4 x+y+2 & =0
\end{aligned}
$