The slope of the line touching both the parabolas y²=4x and x²=-32y is

Equation of tangent at A(t²,2t) to y²=4x is y=x/t + t.
This is tangent to x²=-32y at B ⇒ x² + 32(x/t + t) = 0 ⇒ x² + 32x/t + 32t = 0.
For tangency, the discriminant must be zero: (32/t)² - 4(32t) = 0 ⇒ 32(32/t² - t) = 0 ⇒ t³ = 8 ⇒ t = 2.
Slope of tangent is 1/t = 1/2