If the slope of one of the lines represented by $a x^2-6 x y+$ $y^2=0$ is the square of the other, then the value of $a$ is

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Aicharya · Accepted Answer

Given pair of lines, $a x^2-6 x y+y^2=0$ Let the slope of one line be $m$, then slope of another line will be $m^2$. We know that

On cubing Eq. (i) both sides, we get $\begin{aligned} & \left(m+m^2\right)^3=(6)^3 \\ & \Rightarrow \quad m^3+m^6+3 m^3\left(m+m^2\right)=216 \\ & \Rightarrow \quad m^3+m^6+18 m^3=216 \quad\left[\because m^3=a\right] \end{aligned}$ $\begin{aligned} & \Rightarrow \quad a+a^2+18 a=216 \\ & \Rightarrow \quad a^2+19 a-216=0 \\ & a^2+27 a-8 a-216=0 \\ & a(a+27)-8(a+27)=0 \\ & (a+27)(a-8)=0 \\ & \therefore \quad a=-27,8\end{aligned}$

If the slope of one of the lines represented by $a x^2-6 x y+$ $y^2=0$ is the square of the other, then the…

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