It is given that the equation of the tangent to the given curve is

y = mx + 1

Now, substituting the value of y in y² = 4x, we get

⇒ (mx + 1)² = 4x

⇒ m²x² + 1 + 2mx - 4x =0

⇒ m²x² + x(2m - 4) + 1 = 0………………..(1)

Since, a tangent touches the curve at one point, the root of equation (1) must be equal.

Thus, we get

Discriminant = 0

(2m - 4)² – 4(m²)(1) = 0

⇒ 4m² + 16 - 16m - 4m² =0

⇒ 16 – 16m = 0

⇒ m =1

Therefore, the required value of m is 1.