Two parabolas with a common vertex and with axes along the x -axis...

Equation of two parabola are ${y}^{2} = 3x$ and ${x}^{2} = 3y$.

Let equation of tangent to ${y}^{2} = 3x$ is $y = mx + \dfrac{3}{4m}$ is also tangent to ${x}^{2} = 3y$

$⇒ {x}^{2} = 3mx + \dfrac{9}{4m}$

$⇒ 4m{x}^{2} - 12{m}^{2}x - 9 = 0$ have equal roots

$⇒ D = 0$

$⇒ 144{m}^{4} = 4\left(4m\right) ( - 9 )$

$⇒ {m}^{4} + m = 0 ⇒ m = - 1$

Hence, common tangent is $y = - x - \dfrac{3}{4}$

$⇒ 4\left(x + y\right) + 3 = 0$