A hyperbola whose transverse axis is along the major axis of the conic x²/3 + y²/4 = 4 and has vertices at the foci of this conic. If the eccentricity of the hyperbola is 3/2, then which of the following points does NOT lie on it?

The given conic is x²/3 + y²/4 = 4, which can be written as x²/12 + y²/16 = 1. This is an ellipse with a² = 12 and b² = 16. Thus, a = 2√3 and b = 4. The major axis is along the y-axis.

The eccentricity of the ellipse is e = √(1 - a²/b²) = √(1 - 12/16) = √(1/4) = 1/2.

The foci of the ellipse are at (0, ±be) = (0, ±4(1/2)) = (0, ±2).

The hyperbola has vertices at the foci of the ellipse, which are (0, 2) and (0, -2). The transverse axis of the hyperbola is along the y-axis. The equation of the hyperbola is of the form y²/a² - x²/b² = 1.

The vertices of the hyperbola are at (0, ±a), which are (0, ±2). Therefore, a = 2.

The eccentricity of the hyperbola is given as 3/2. The eccentricity of a hyperbola is e = √(1 + b²/a²). We have:

(3/2)² = 1 + b²/2²

9/4 = 1 + b²/4

5/4 = b²/4

b² = 5

Therefore, the equation of the hyperbola is y²/4 - x²/5 = 1.

Now let's check the given points:

(0, 2): 2²/4 - 0²/5 = 1, which is true.
(√10, 2√3): (2√3)²/4 - (√10)²/5 = 12/4 - 10/5 = 3 - 2 = 1, which is true.
(√5, 2√2): (2√2)²/4 - (√5)²/5 = 8/4 - 5/5 = 2 - 1 = 1, which is true.
(5, 2√3): (2√3)²/4 - 5²/5 = 12/4 - 25/5 = 3 - 5 = -2 ≠ 1, which is false.

Therefore, the point (5, 2√3) does NOT lie on the hyperbola.