To determine the coordinates of the focus and the equation of the directrix for the parabola described by the equation [tex]\( x^2 = 2y \)[/tex], let's follow these steps:
1. Standard Form of the Parabola Equation:
The standard form for a vertical parabola is [tex]\((x - h)^2 = 4p(y - k)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex, [tex]\( p \)[/tex] is the distance from the vertex to the focus, and the directrix is [tex]\( y = k - p \)[/tex].
2. Identifying the Parameters:
- Compare the given equation [tex]\( x^2 = 2y \)[/tex] with the standard form [tex]\( (x - 0)^2 = 4p(y - 0) \)[/tex].
- Here, [tex]\( h = 0 \)[/tex], [tex]\( k = 0 \)[/tex], and [tex]\( 4p = 2 \)[/tex].
3. Solving for [tex]\( p \)[/tex]:
- From [tex]\( 4p = 2 \)[/tex], we find [tex]\( p = \frac{1}{2} \)[/tex].
4. Finding the Focus:
- The focus is at [tex]\( (h, k + p) \)[/tex], which substitutes to [tex]\( (0, 0 + \frac{1}{2}) \)[/tex]. Hence, the focus is [tex]\( \left(0, \frac{1}{2}\right) \)[/tex].
5. Finding the Directrix:
- The directrix is given by the equation [tex]\( y = k - p \)[/tex], which substituting values becomes [tex]\( y = 0 - \frac{1}{2} \)[/tex]. Therefore, the equation of the directrix is [tex]\( y = -\frac{1}{2} \)[/tex].
So, the coordinates of the focus and the equation of the directrix are:
- Focus: [tex]\(\left(0, \frac{1}{2}\right)\)[/tex]
- Directrix: [tex]\( y = -\frac{1}{2} \)[/tex]
The correct answer is:
- Focus: [tex]\(\left(0, \frac{1}{2}\right)\)[/tex]; Directrix: [tex]\( y = -\frac{1}{2} \)[/tex]
