Question

Find the equation of the parabola which is symmetric about the $y$ - axis and passes through the point $(2,–3)$.

Answer 1

Given: Parabola is symmetrix about $y$ - axis and passes through point $(2,-3).$
Step $1$ : Finding $a$
Since the parabola is symmetric about $y$ − axis and has its vertex at the origin, the equation is of the form $x^2=4ay$ or $x^2=-4ay.$
Since $(2,-3)$ lies in the fourth quadrant, so the parabola will be of the form $x^2=-4ay$
Now parabola passes through $(2,-3)$
Putting $x=2,y=-3$ in equation
$\Rightarrow 2^2=-4\times a\times -3$
$\Rightarrow 4=12a$
$\Rightarrow a=\frac{4}{12}$
$\Rightarrow a=\frac{1}{3}$
$a=\frac{1}{3}$

Step $2$: Equation of parabola is
$\Rightarrow x^2=-4ay$
$\Rightarrow x^2=-\frac{4}{3}y$