If the equations 4x + (k − 2)y + 3 = 0 and (k − 2)x + 9y − 5 = 0 (k > 0) are…

Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel if the ratios of their coefficients are equal:
a₁ / a₂ = b₁ / b₂

Step-by-Step Calculation
Identify coefficients:

a₁ = 4, b₁ = (k - 2)

a₂ = (k - 2), b₂ = 9

Set up the ratio:
4 / (k - 2) = (k - 2) / 9

Cross-multiply:
(k - 2)² = 4 × 9
(k - 2)² = 36

Solve for k:
k - 2 = ±√36
k - 2 = 6 or k - 2 = -6
k = 8 or k = -4

Since the problem states k > 0, we take k = 8.

Calculate k² + 6:
(8)² + 6 = 64 + 6 = 70