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 \(x^2 = -k y\) (with \(k > 0\)):
- The focus is at \(\left(0, -\frac{k}{4}\right)\).
- The directrix is at \(y = \frac{k}{4}\).
Here, \(k = \frac{25}{2}\), so the directrix is \(y = \frac{25/2}{4} = \frac{25}{8}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the equation of a parabola: \(x^2 = -\frac{25}{2}y\).
We need to find the equation of its directrix.
Key Formula or Approach:
The standard equation of a downward-opening vertical parabola is:
\[ x^2 = -4ay \quad (\text{where } a > 0) \]
For this standard parabola: - The vertex is at \((0,0)\).
- The focus is at \((0, -a)\).
- The directrix is a horizontal line given by the equation:
\[ y = a \]

Step 2: Detailed Explanation:

• Compare the given equation of the parabola with the standard form:
Given: \(x^2 = -\frac{25}{2}y\)
Standard: \(x^2 = -4ay\)

• Equate the coefficients of \(y\) on both sides:
\[ -4a = -\frac{25}{2} \] \[ 4a = \frac{25}{2} \]

• Solve for \(a\):
\[ a = \frac{25}{8} \]

• The directrix of the downward-opening parabola is given by:
\[ y = a \] Therefore:
\[ y = \frac{25}{8} \]

Step 3: Final Answer:
The equation of its directrix is \(y = \frac{25}{8}\).