The equation of tangent to the curve $y=\sqrt{3x-2}$ which is parallel to the line $4x-2y+5=0$ is:

$48x-24y=23$

The correct answer is Option (4) → $48-24y=23$

curve: $y=\sqrt{3x-2}$

line: $4x-2y+5=0$

curve || line

so $y=2x+\frac{5}{2}$

$\frac{dy}{dx}=2$ slope of line

Slope at same point = slope of line

so $y^2=3x-2$

so $(2y\frac{dy}{dx}=3)=2$ slope of line

$⇒\frac{3}{2y}=2$ so $y=\frac{3}{4}$ as $y=\sqrt{3x-2}$

so $x=\frac{y^2+2}{3}$

$⇒x=\frac{\frac{9}{16}+2}{3}=\frac{41}{48}$

so equation of tangent

$(y-\frac{3}{4})=2(x-\frac{41}{48})$

$48y-36=96x-82$

$96x-48y=46$

$⇒48x-24y=23$