Question

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

Answer 1

One of the lines represented by x² + 2xy + 4y + k = 0 is x + 2 = 0.       .....(1)

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,

${\displaystyle \frac{\text{a}}{1}=\frac{\text{b}}{2}=\frac{\text{2b}}{4}=\frac{\text{2c}}{\text{k}}}$ and 2a + c = 0

∴ a = ${\displaystyle \frac{\text{2c}}{\text{k}}}$ and c = - 2a

∴ a = ${\displaystyle \frac{2(-2\text{a})}{\text{k}}}$

∴ ${\displaystyle 1=\frac{-4}{\text{k}}}$

∴ k = - 4