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The point of intersection of the ellipse x²/16 + y²/b² = 1 and the curve y² = 3x² lies on both.
Substitute y² = 3x² into the ellipse equation:
x²/16 + 3x²/b² = 1
x² (1/16 + 3/b²) = 1
x² (b² + 48) / 16b² = 1
x² = 16b² / (b² + 48).
For a solution to exist, we need x² > 0, which is true if b≠0.
The problem seems to have a condition missing or misinterpreted in the OCR. The provided solution also shows x² + y² = 4b, which might be another curve involved. Assuming the point lies on x²+y²=4b.

x² + 3x² = 4b => 4x² = 4b => x² = b.
Substitute x²=b into the ellipse equation: b/16 + 3b/b² = 1 (assuming y²=3b).
b/16 + 3/b = 1
b² + 48 = 16b
b² - 16b + 48 = 0
(b-12) (b-4) = 0
So b = 12 or b = 4. Given b>4, we have b=12.

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Given r x a = b x r, which means r x a + r x b = 0, so r x (a+b) = 0.
This implies r is parallel to (a+b). So, r = λ (a+b).
a = I + 2j - 3k, b = 2i - 3j + 5k
a+b = 3i - j + 2k.
r = λ (3i - j + 2k).
Given r ⋅ (αi + 2j + k) = 3. The OCR is unclear, but the equation appears to be r ⋅ (αi + 2j + k) = 3. The solution works with r ⋅ (αi + 2j + k) = 3, but theOCR says r. (ai+2j+k). Let's assume it's α.
λ (3i - j + 2k) ⋅ (αi + 2j + k) = 3
λ (3α - 2 + 2) = 3 => λα = 1.
Given r ⋅ (2i + 5j - αk) = -1.
λ (3i - j + 2k) ⋅ (2i + 5j - αk) = -1
λ (6 - 5 - 2α) = -1 => λ (1 - 2α) = -1.
From λα = 1, α = 1/λ.
λ (1 - 2/λ) = -1 => λ - 2 = -1 => λ = 1.
If λ = 1, then α = 1.
r = 1 * (3i - j + 2k) = 3i - j + 2k.
|r| = √ (3² + (-1)² + 2²) = √ (9+1+4) = √14.
We need to find α + |r|².
α + |r|² = 1 + (√14)² = 1 + 14 = 15.

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