Step-by-Step Solution

1. The direction ratios of the first line are $\alpha, -5, \beta$ and the direction ratios of the second line are $1, 0, 1$.

2. The angle between the lines is given by $\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}}$.

3. Here, $\theta = \frac{\pi}{4}$, so $\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}$.

4. Substituting the direction ratios, we get $\frac{1}{\sqrt{2}} = \frac{\alpha(1) + (-5)(0) + \beta(1)}{\sqrt{\alpha^2 + (-5)^2 + \beta^2}\sqrt{1^2 + 0^2 + 1^2}}$.

5. Simplifying, $\frac{1}{\sqrt{2}} = \frac{\alpha + \beta}{\sqrt{\alpha^2 + 25 + \beta^2}\sqrt{2}}$.

6. Further simplification gives $1 = \frac{\alpha + \beta}{\sqrt{\alpha^2 + 25 + \beta^2}}$.

7. Squaring both sides, we get $1 = \frac{(\alpha + \beta)^2}{\alpha^2 + 25 + \beta^2}$.

8. Therefore, $\alpha^2 + 25 + \beta^2 = (\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2$.

9. This simplifies to $25 = 2\alpha\beta$.

10. Hence, the relation between $\alpha$ and $\beta$ is $\alpha\beta = \frac{25}{2}$.