Find the equation of the parabola which passes through ( 6 , - 2 ) , has its vertex at the origin and its…

As given, the vertex of the parabola is origin and it's axis is along the $y-$axis,

So, the standard equation of the parabola with given conditions is, $x^2=-4ay ...\left(1\right)$, as it passes through $\left(6, -2\right)$

Hence, $x^2=-4ay \Rightarrow {\left(6\right)}^2=-4\times a\times \left(-2\right) \Rightarrow a=\frac{36}{8}=\frac{9}{2}$

So, substituting the value of $a$ in equation $\left(1\right)$

The equation is $x^2=-18y$