Question

If line x + 2 = 0 coincides with one of the lines represented by the equation x² + 2xy + 4y + k = 0 then show that k = -4.

Answer 1

One of the lines represented by

x² + 2xy + 4y + k = 0 … (1)

is x + 2 = 0.

Let the other line represented by (1) be ax + by + c = 0.

∴ their combined equation is (x + 2)(ax + by + c) = 0

∴ ax² + bxy + cx + 2ax + 2by + 2c = 0

∴ ax² + bxy + (2a + c)x + 2by + 2c — 0 … (2)

As the equations (1) and (2) are the combined equations of the same two lines, they are identical.

∴ by comparing their corresponding coefficients, we get,