Question

If a ≠ b ≠ c, write the condition for which the equations (b − c) x + (c − a) y + (a − b) = 0 and (b³ − c³) x + (c³ − a³) y + (a³ − b³) = 0 represent the same line.

Answer 1

The given lines are

(b − c)x + (c − a)y + (a − b) = 0 ... (1)

(b³ − c³)x + (c³ − a³)y + (a³ − b³) = 0 ... (2)

The lines (1) and (2) will represent the same lines if

$$\begin{gathered} \frac{b-c}{b^3-c^3}=\frac{c-a}{c^3-a^3}=\frac{a-b}{a^3-b^3} \\ \Rightarrow \frac{b-c}{\left(b-c\right)\left(b^2+bc+c^2\right)}=\frac{c-a}{\left(c-a\right)\left(c^2+ac+a^2\right)}=\frac{a-b}{\left(a-b\right)\left(a^2+ab+b^2\right)} \\ \Rightarrow \frac{1}{b^2+bc+c^2}=\frac{1}{c^2+ac+a^2}=\frac{1}{a^2+ab+b^2}\left(\because a\ne b\ne c\right) \end{gathered}$$

$$\begin{gathered} \Rightarrow b^2+bc+c^2=c^2+ac+a^2\mathrm{and}{c}^2+ac+a^2=a^2+ab+b^2 \\ \Rightarrow \left(a-b\right)\left(a+b+c\right)=0\mathrm{and}\left(b-c\right)\left(b+c+a\right)=0 \\ \Rightarrow a+b+c=0\left(\because a\ne b\ne c\right) \end{gathered}$$

Hence, the given lines will represent the same lines if a + b + c = 0.