The square of the distance of the point of intersection of the lines $$\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})$$, $$a \neq 0$$ and $$\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})$$ from the origin is :

Any general point on Line 1 has the coordinates: $$R_1 = (1 + a\lambda, \, 1 - \lambda, \, -1)$$

Any general point on Line 2 has the coordinates: $$R_2 = (4 + 2\mu, \, 0, \, -1 + a\mu)$$

Since the lines intersect, $$R_1$$ must equal $$R_2$$ component-by-component:

$$1 - \lambda = 0 \implies \lambda = 1$$

$$-1 = -1 + a\mu \implies a\mu = 0$$. Since $$a \neq 0$$ $$\implies$$ $$\mu = 0$$

$$1 + a\lambda = 4 + 2\mu$$

$$1 + a(1) = 4 + 2(0)$$

$$1 + a = 4 \implies a = 3$$

$$\text{Point of intersection } (x, y, z) = (4 + 2(0), \, 0, \, -1 + 3(0)) = (4, \, 0, \, -1)$$

$$OR^2 = x^2 + y^2 + z^2$$

$$OR^2 = 4^2 + 0^2 + (-1)^2$$

$$OR^2 = 16 + 0 + 1 = 17$$