Question

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

• A. $x^2=4y$
• B. $4y=3x^2$
• C. $3x^2=-4y$
• D. $3y=-4x^2$

Answer 1

Answer: (C)

Since the parabola is symmetric about $y$-axis and has its vertex at the origin so the equation is of the form $x^2=4ay$ or $x^2=-4ay$, where the sign depends on whether the parabola opens upwards or downwards. But the parabola passes through $(2,-3)$ which lies in the fourth quadrant so it must open downwards. Thus the equation is of the form $x^2=-4ay$.
Since the parabola passes through $(2,-3)$, we have
$2^2=-4a(-3)$,
i.e., $a=\frac{1}{3}$
Therefore, the equation of the parabola is
$x^2=-4\left(\frac{1}{3}\right)y$,
i.e., $3x^2=-4y$.