Question

The equation of the Parabola, whose focus is $(0, -2)$ and the vertex is $(0, 0)$, is

• A. $y^2 = 32x$
• B. $x^2 = -8y$
• C. $x^2 = 4y$
• D. $y^2 = -8x$

Hint:

If the focus has a non-zero $y$-coordinate and a zero $x$-coordinate, the equation must start with $x^2$. If the $y$-coordinate is negative, the coefficient on the right side must be negative. This immediately points to $x^2 = -4ay$.

Answer 1

The Correct Option is B

To find the equation, we determine the orientation of the parabola based on the position of the vertex and focus. 1. Determine Orientation: Vertex is at $(0, 0)$. Focus is at $(0, -2)$. Since the focus lies on the negative $y$-axis, the parabola opens downwards. The standard form for a downward-opening parabola with vertex at origin is: $$x^2 = -4ay$$

2. Find the value of 'a': The value of $a$ is the distance between the vertex and the focus. $$a = \text{Distance between } (0, 0) \text{ and } (0, -2) = 2$$

3. Form the Equation: Substitute $a = 2$ into the standard form: $$x^2 = -4(2)y$$ $$x^2 = -8y$$ Option (B) is the correct representation.