The locus of the point of intersection of the straight lines tx - y - t = 0, x - ty + 3 = 0 (t ∈ R), is.

tx - y - t = 0 and x - ty + 3 = 0
From the second equation, we have t = (x + 3) / 2y
Substituting this value in the first equation, we have (x - y) * (x + 3) / 2y - y = 0
=> x² - y² + 3x - 5y = 0
=> x² + 3x - (y² + 5y) = 0
=> x² + 3x + 9/4 - (y² + 5y + 25/4) = 0 - 9/4 + 25/4
=> (x + 3/2)² - (y + 5/2)² = 16/4 = 4
=> (x + 3/2)² / 4 - (y + 5/2)² / 4 = 1
This is a hyperbola.
Eccentricity = √(1 + b²/a²) = √(1 + 1) = √2
Also, semi conjugate axis = √4 = 2
Therefore, the length of conjugate axis would be 4.