The line y=m x+1 is a tangent to the curve

Solution:

It is given that the line $y=m x+1$ is a tangent to the curve $y^{2}=4 x$.

Slope of the line $=\mathrm{m}$

Slope of the curve $\frac{d y}{d x}$,

Differentiating the curve we get

$2 y \frac{d y}{d x}=4$

$\Rightarrow \frac{d y}{d x}=\frac{2}{y}$

$\Rightarrow \frac{2}{y}=m$

$\Rightarrow y=\frac{2}{m}$

$\because$ The given line is a tangent to the curve so the point passes through both line and curve.

$\Rightarrow y=m x+1$ and $y^{2}=4 x$

$\Rightarrow \frac{2}{m}=m x+1$ and $\frac{4}{m^{2}}=4 x$

$\Rightarrow m x=\frac{2-m}{m}$ and $x=\frac{1}{m^{2}}$

$\Rightarrow x=\frac{2-m}{m^{2}}$ and $x=\frac{1}{m^{2}}$

$\Rightarrow \frac{2-m}{m^{2}}=\frac{1}{m^{2}}$

$\Rightarrow 2-m=1$

$\Rightarrow m=1$

Hence, the correct option is A.