Question

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

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

Hint:

For parabolas symmetric about the y-axis, always assume $x^2 = ky$ unless vertex shift is mentioned.

Answer 1

The Correct Option is D

Step 1: Substitute the point into general form Given point: \[ (2, -3) \] Substitute into $x^2 = ky$: \[ (B)^2 = k(-3) \] \[ 4 = -3k \]

Step 2: Solve for constant $k$ \[ k = -\frac{4}{3} \] This negative value indicates that the parabola opens downward.

Step 3: Form the equation \[ x^2 = -\frac{4}{3}y \] Multiply throughout by 3 to remove fraction: \[ 3x^2 = -4y \]

Step 4: Geometrical interpretation Since $k < 0$, the parabola opens in the negative y-direction, which is consistent with the given point $(2,-3)$ lying below the x-axis. Final Answer: \[ {3x^2 = -4y} \]