Answer :
To determine the coordinates of the focus of the given parabola [tex]\( x^2 = 4y \)[/tex], let's first recall the standard form of a parabola that opens upwards or downwards.
The general form of a parabola that opens upwards or downwards is:
[tex]\[ x^2 = 4ay \][/tex]
In this equation, [tex]\(a\)[/tex] represents the distance from the vertex to the focus. The vertex of this parabola is at the origin [tex]\((0,0)\)[/tex].
Given the equation:
[tex]\[ x^2 = 4y \][/tex]
We compare this with the general form [tex]\( x^2 = 4ay \)[/tex]. By comparing, we can see that [tex]\( 4a = 4 \)[/tex].
Solving for [tex]\( a \)[/tex]:
[tex]\[ 4a = 4 \][/tex]
[tex]\[ a = 1 \][/tex]
Therefore, for the given parabola [tex]\( x^2 = 4y \)[/tex], the distance from the vertex to the focus is [tex]\( a = 1 \)[/tex]. Since the parabola opens upwards and the vertex is at the origin [tex]\((0, 0)\)[/tex], the focus will be located at [tex]\((0, a)\)[/tex] which is [tex]\((0, 1)\)[/tex].
Thus, the coordinates of the focus of the parabola are:
[tex]\((0, 1)\)[/tex]
Therefore, the correct option is:
[tex]\((0, 1)\)[/tex]
The general form of a parabola that opens upwards or downwards is:
[tex]\[ x^2 = 4ay \][/tex]
In this equation, [tex]\(a\)[/tex] represents the distance from the vertex to the focus. The vertex of this parabola is at the origin [tex]\((0,0)\)[/tex].
Given the equation:
[tex]\[ x^2 = 4y \][/tex]
We compare this with the general form [tex]\( x^2 = 4ay \)[/tex]. By comparing, we can see that [tex]\( 4a = 4 \)[/tex].
Solving for [tex]\( a \)[/tex]:
[tex]\[ 4a = 4 \][/tex]
[tex]\[ a = 1 \][/tex]
Therefore, for the given parabola [tex]\( x^2 = 4y \)[/tex], the distance from the vertex to the focus is [tex]\( a = 1 \)[/tex]. Since the parabola opens upwards and the vertex is at the origin [tex]\((0, 0)\)[/tex], the focus will be located at [tex]\((0, a)\)[/tex] which is [tex]\((0, 1)\)[/tex].
Thus, the coordinates of the focus of the parabola are:
[tex]\((0, 1)\)[/tex]
Therefore, the correct option is:
[tex]\((0, 1)\)[/tex]
