Answer :
To determine the coordinates of the focus and the equation of the directrix for the given parabola [tex]\( x^2 = 2y \)[/tex], we can rewrite the given equation into the standard form of a parabola. The general form for a parabola that opens upwards or downwards and has its vertex at the origin [tex]\((0,0)\)[/tex] is [tex]\( x^2 = 4py \)[/tex].
Here are the steps:
1. Rewrite the equation in standard form:
Given:
[tex]\[ x^2 = 2y \][/tex]
We need to compare this with the standard form [tex]\( x^2 = 4py \)[/tex]. By inspection, we can see:
[tex]\[ 4p = 2 \][/tex]
To find [tex]\( p \)[/tex], we solve for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{2}{4} = \frac{1}{2} \][/tex]
2. Determine the coordinates of the focus:
For a parabola in the form [tex]\( x^2 = 4py \)[/tex], the focus is at [tex]\((0, p)\)[/tex].
With [tex]\( p = \frac{1}{2} \)[/tex], the coordinates of the focus are:
[tex]\[ (0, \frac{1}{2}) \][/tex]
3. Determine the equation of the directrix:
The directrix of a parabola [tex]\( x^2 = 4py \)[/tex] is given by:
[tex]\[ y = -p \][/tex]
With [tex]\( p = \frac{1}{2} \)[/tex], the equation of the directrix is:
[tex]\[ y = -\frac{1}{2} \][/tex]
Combining these results, we have:
- The coordinates of the focus are [tex]\((0, \frac{1}{2})\)[/tex].
- The equation of the directrix is [tex]\( y = -\frac{1}{2} \)[/tex].
Thus, the correct choice is:
[tex]\[ \text{focus: } \left(0, \frac{1}{2}\right); \text{ directrix: } y=-\frac{1}{2} \][/tex]
Here are the steps:
1. Rewrite the equation in standard form:
Given:
[tex]\[ x^2 = 2y \][/tex]
We need to compare this with the standard form [tex]\( x^2 = 4py \)[/tex]. By inspection, we can see:
[tex]\[ 4p = 2 \][/tex]
To find [tex]\( p \)[/tex], we solve for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{2}{4} = \frac{1}{2} \][/tex]
2. Determine the coordinates of the focus:
For a parabola in the form [tex]\( x^2 = 4py \)[/tex], the focus is at [tex]\((0, p)\)[/tex].
With [tex]\( p = \frac{1}{2} \)[/tex], the coordinates of the focus are:
[tex]\[ (0, \frac{1}{2}) \][/tex]
3. Determine the equation of the directrix:
The directrix of a parabola [tex]\( x^2 = 4py \)[/tex] is given by:
[tex]\[ y = -p \][/tex]
With [tex]\( p = \frac{1}{2} \)[/tex], the equation of the directrix is:
[tex]\[ y = -\frac{1}{2} \][/tex]
Combining these results, we have:
- The coordinates of the focus are [tex]\((0, \frac{1}{2})\)[/tex].
- The equation of the directrix is [tex]\( y = -\frac{1}{2} \)[/tex].
Thus, the correct choice is:
[tex]\[ \text{focus: } \left(0, \frac{1}{2}\right); \text{ directrix: } y=-\frac{1}{2} \][/tex]
