Question

Let \(x^2 = -\frac{25}{2}y\) be the equation of a parabola, then the equation of its directrix is given by :

• A. \(y = \frac{25}{2}\)
• B. \(y = -\frac{25}{4}\)
• C. \(y = -\frac{25}{8}\)
• D. \(y = \frac{25}{8}\)

Hint:

For any parabola of the form \(x^2 = \pm k y\):
The distance from the vertex to the focus or directrix is always:
\[ a = \frac{k}{4} \]
Here, \(k = \frac{25}{2}\), so \(a = \frac{25/2}{4} = \frac{25}{8}\).
Since the parabola opens downwards (\(-\) sign), the directrix lies above the x-axis, so its equation is positive: \(y = \frac{25}{8}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To find the directrix of a parabola, we compare the given equation with the standard forms of a parabola.
A parabola with its vertex at the origin and symmetric about the y-axis has the standard equation:
\[ x^2 = -4ay \quad (a > 0) \]
This parabola opens downwards, its focus is at \((0, -a)\), and its directrix is the horizontal line \(y = a\).

Step 2: Detailed Explanation:
Let us compare the given equation of the parabola with the standard form:
Given equation:
\[ x^2 = -\frac{25}{2}y \]
Standard downward-opening equation:
\[ x^2 = -4ay \]
Equating the coefficients of \(y\):
\[ 4a = \frac{25}{2} \]
To find the value of \(a\), divide both sides by 4:
\[ a = \frac{25}{2 \times 4} = \frac{25}{8} \]
For a downward-opening parabola \(x^2 = -4ay\):
- The vertex is at \((0, 0)\).
- The focus is at \((0, -a) = \left(0, -\frac{25}{8}\right)\).
- The directrix is a line perpendicular to the axis of symmetry, located at an equal distance \(a\) on the opposite side of the vertex.
Therefore, the equation of the directrix is:
\[ y = a \implies y = \frac{25}{8} \]
This matches Option (D).

Step 3: Final Answer:
The correct option is (D).