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Explanation:

AE is altitude from A on BC. ∠AEC = 90°
It is given,

∠ABC = 60°, ∠ACB = 45°

In triangle ABE,
AE = $\sqrt{3}$x
As ∠EAC = ∠ECA , EC = $\sqrt{3}$x
In triangle AED, ∠EAD = 60°
Therefore, CD = 3x - $\sqrt{3}$x
It is given,

BC ∙ CD = $\frac{3^4.2^3.11}{19}$

x = ($\sqrt{3}$ + 1)$\sqrt{3}$($\sqrt{3}$ - 1)x = $\frac{3^4.2^3.11}{19}$

x = $\frac{3^4.2^3.11}{19}$

AE = 9*2 = 18
AE² = 324
Answer is option B.