Conditions for Homogeneity/Straight Line

Theorem
The line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ is homogeneous $\beta = 0$.

Proof
Let the line $L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$ be homogeneous.

Then the origin $\tuple {0, 0}$ lies on the line $L$.

That is, $\alpha_1 0 + \alpha_2 0 = \beta \implies \beta = 0$.

Let the equation of $L$ be $L = \alpha_1 x_1 + \alpha_2 x_2 = 0$.

Then $0 = \alpha_1 0 + \alpha_2 0 \in L$ and so $\tuple {0, 0}$ lies on the line $L$.

Hence $L$ is homogeneous.