If the line x - 2y = 12 is tangent to the ellipse x²/a² + y²/b² = 1 at the point (3, -9/2), then the length of the latus rectum of the ellipse is:

The correct option is A

Tangent at (3, -9/2)
The equation of the tangent to the ellipse x²/a² + y²/b² = 1 at the point (x₁, y₁) is given by xx₁/a² + yy₁/b² = 1.
Given that the line x - 2y = 12 is tangent to the ellipse x²/a² + y²/b² = 1 at the point (3, -9/2), we have:
3x/a² - (9/2)y/b² = 1
x - 2y = 12
Comparing the two equations, we get:
3/a² = k
-9/(2b²) = -2k
where k is a constant.
From the second equation, we have 9/(2b²) = 2k, which implies k = 9/(4b²).
Therefore, 3/a² = 9/(4b²), which simplifies to 4b² = 3a².
Since the point (3, -9/2) lies on the ellipse, we have:
(3)²/a² + (-9/2)²/b² = 1
9/a² + 81/(4b²) = 1
Substituting 4b² = 3a², we get:
9/a² + 81/(3a²) = 1
9/a² + 27/a² = 1
36/a² = 1
a² = 36
Then, b² = 3a²/4 = 3(36)/4 = 27
The length of the latus rectum of the ellipse is given by 2b²/a = 2(27)/6 = 9.