Question

The equation $a{x}^3-9x^{2}y-x{y}^2+4y^3=0$, $(a>0)$ represents three straight lines, two of which are perpendicular to each other, then the value of $a$ is

Answer 1

$a{x}^3-9x^{2}y-x{y}^2+4y^3=0$
Let assume, $\frac{y}{x}=t,$
$4t^3-t^2-9t+a=0$......... {1}
Two lines are perpendicular to each other, then their slopes $m_1,m_2$ satisfies the condition $m_1{m}_2=-1\Rightarrow m_2=-\frac{1}{m_1}$.
Roots of the above equation gives us the slopes of the three lines, i.e. $m_1,m_2,$ and $m_3$,
Now, we have $m_1{m}_2{m}_3=-\frac{a}{4}\Rightarrow m_3=\frac{a}{4}$
$m_3$ is a root of equation {1},
$\therefore a(a^2-a-20)=0\Rightarrow a=\{-4,0,5\}$
$\because a>0$ (given), $a=5$.