An ellipse passes through the foci of the hyperbola 9x² - 4y² = 36. Its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is 1/2, then which of the following points does not lie on the ellipse?

Given equation of hyperbola is 9x² - 4y² = 36 ⟹ x²/4 - y²/9 = 1
Here, a = 2, b = 3
e' = √(1 + b²/a²) = √(1 + 9/4) = √13/2
The foci of the hyperbola are (±ae', 0) = (±2(√13/2), 0) = (±√13, 0)
Let the equation of the ellipse be x²/A² + y²/B² = 1
Since the ellipse passes through the foci of the hyperbola, it passes through (±√13, 0).
Therefore, 13/A² = 1 ⟹ A² = 13
The product of eccentricities is given as 1/2.
Let e be the eccentricity of the ellipse.
Then, e'e = 1/2
√(1 - B²/A²) (√13/2) = 1/2
√(1 - B²/13) = 1/√13
1 - B²/13 = 1/13
B²/13 = 12/13
B² = 12
The equation of the ellipse is x²/13 + y²/12 = 1
Now let's check which point does not satisfy this equation:
(1) (12√13, √32): (144(13)/13) + (32/12) = 144 + 8/3 ≠ 1
(2) (√13, 0): 13/13 + 0 = 1
(3) (√132, √6): 132/13 + 6/12 = 10.15 + 0.5 ≠ 1
(4) (√392, √3): 39/13 + 3/12 = 3 + 1/4 ≠ 1
Thus, points (1), (3), and (4) do not lie on the ellipse.