Question

If the three lines ax + a²y + 1 = 0, bx + b²y + 1 = 0 and cx + c²y + 1 = 0 are concurrent, show that at least two of three constants a, b, c are equal.

Answer 1

The given lines can be written as follows:
ax + a²y + 1 = 0           ... (1)
bx + b²y + 1 = 0           ... (2)
cx + c²y + 1 = 0           ... (3)
The given lines are concurrent. 

\[\therefore \begin{vmatrix}a & a^2 & 1 \\ b & b^2 & 1 \\ c & c^2 & 1\end{vmatrix} = 0\]

Applying the transformation \[R_1 \to R_1 - R_2\text{  and } R_2 \to R_2 - R_3\]:

\[\begin{vmatrix}a - b & a^2 - b^2 & 0 \\ b - c & b^2 - c^2 & 0 \\ c & c^2 & 1\end{vmatrix} = 0\]

\[ \Rightarrow \left( a - b \right)\left( b - c \right)\begin{vmatrix}1 & a + b & 0 \\ 1 & b + c & 0 \\ c & c^2 & 1\end{vmatrix} = 0\]

\[ \Rightarrow \left( a - b \right)\left( b - c \right)\left( c - a \right) = 0\]

\[\Rightarrow a - b = 0 \text { or b - c = 0 or c - a} = 0\]

\[ \Rightarrow \text { a = b or b = c or c } = a\]

Therefore, atleast two of the constants a,b,c are equal .