Find the coordinates of the point on the curve

Given that the curve $y^{2}=3-4 x$ has a point where tangent is $\|$ to the line $2 x+y-2=0$.

Slope of the given line is $-2$

$\because$ the point lies on the curve

$\therefore y^{2}=3-4 x$

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

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

Now, the slope of the curve $=$ slope of the line

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

$\Rightarrow y=1$

Putting above value in the equation of the line,

$2 x+1-2=0$

$\Rightarrow 2 x-1=0$

$\Rightarrow x=\frac{1}{2}$

So, the required coordinate is $\left(\frac{1}{2}, 1\right)$.