Let $L$ be the line joining the origin to the point of intersection of the lines represented by $2 x^2-3 x…
Let $L$ be the line joining the origin to the point of intersection of the lines represented by $2 x^2-3 x y$ $-2 y^2+10 x+5 y=0$. If $L$ is perpendicular to the line $k x+y+3=0$, then $k$ is equal to
$\frac {1}{2}$
$\frac {-1}{2}$
$-1$
$\frac {1}{3}$
Solution
Given that,
$\begin{aligned}
& 2 x^2-3 x y-2 y^2+10 x+5 y=0 \\
& (2 x+y)(x-2 y+5)=0 \\
& 2 x+y=0 \text { and } x-2 y+5=0
\end{aligned}$
Now, equation of line passing through origin is
$2 x^2+y=0 \quad \Rightarrow m_1=-2$
Since, this line is perpendicular to the line
$\begin{aligned}
& k x+y+3=0 \Rightarrow m_2=-k \\
& \therefore \quad m_1 \times m_2=-1 \\
& \therefore \quad(-2)+(-\mathrm{k})=-1 \quad \Rightarrow k=-\frac{1}{2}
\end{aligned}$