Question

Find the separate equation of the lines represented by the following equation :

(x – 2)² – 3(x – 2)(y + 1) + 2(y + 1)² = 0

Answer 1

(x – 2)² – 3(x – 2)(y + 1) + 2(y + 1)2 = 0

∴ (x – 2)² – 2(x – 2)(y + 1) – (x – 2)(y + 1) + 2(y + 1)² = 0

∴ (x – 2) [(x – 2) – 2(y + 1)] – (y + 1)[(x – 2) – 2(y + 1)] = 0

∴ (x – 2)(x – 2 – 2y – 2) – (y + 1)(x – 2 – 2y – 2) = 0

∴ (x – 2)(x – 2y – 4) – (y + 1)(x – 2y – 4) = 0

∴ (x – 2y – 4)(x – 2 – y – 1) = 0

∴ (x – 2y – 4)(x – y – 3) = 0

∴ the separate equations of the lines are

x – 2y – 4 = 0 and x – y – 3 = 0.

Alternative Method :

(x – 2)² – 3(x – 2)(y + 1) + 2(y + 1)² = 0 … (1)

Put x – 2 = X and y + 1 = Y

∴ (1) becomes,

X² – 3XY + 2Y² = 0

∴ X² – 2XY – XY + 2Y² = 0

∴ X(X – 2Y) – Y(X – 2Y) = 0

∴ (X – 2Y)(X – Y) = 0

∴ the separate equations of the lines are

∴ X – 2Y = 0 and X – Y = 0

∴ (x – 2) – 2(y + 1) = 0 and (x – 2) – (y +1) = 0

∴ x – 2y – 4 = 0 and x – y – 3 = 0.