The shortest distance between the parabolas y^2=4x and y^2=2x-6 is

Hello student,
The given parabolas are y² = 4x and y² = 2x – 6 is
Normal to y² = 4x at (m², 2m) is y + mx – 2m – m³ = 0
Normal to y² = 2x – 6 = 2(x-3) at (m²/2 + 3, m) is y + (x-3) – m – m³/2 = 0
i.e. y + mx – 4m – m³/2 = 0
Now, for both to be coomon or same , we have
– 2m – m³ = – 4m – m³/2
This gives, m(m-2)(m+2) = 0
This gives m = 0, 2, -2.
So, points will be (4, 4) (5, 2).
Hence, the shoretst distance = √1² + 2² = √5 units.