A parabola can be represented by the equation x2 = 2y. What are the coordinates of the focus and the equation of the directrix?

The standard equation for a parabola that opens upward parallel to the y-axis is x2=4ay, where a = distance from the vertex to focus. Vertex is at origin(0,0)
4a=2
a=2/4
a=1/2 or 0.5
the coordinate of the focus is (0, 1/2)
the equation of the directrix is y=-1/2

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MrRoyal MrRoyal · Answer 2 · 2022-01-31T17:49:13+01:00

The coordinate of the focus is (0,1/2) and the equation of the directrix is [tex]y=-\frac 12[/tex]

The equation of the parabola is given as:

[tex]x^2 = 2y[/tex]

A parabola that opens upward parallel to the y-axis is represented as:

[tex]x^2=4ay[/tex]

From the above equation

a represents the distance from the vertex to focus.

And the vertex is at (0,0)

By comparison, we have:

[tex]4a=2[/tex]

Divide through b 4

[tex]a=\frac 12[/tex]

The coordinate of the focus is then calculated as:

[tex]Focus = (0,a)[/tex]

[tex]Focus = (0,\frac 12)[/tex]

The equation of the directrix is calculated as:

[tex]y=-a[/tex]

So, we have:

[tex]y=-\frac 12[/tex]

Hence, the equation of the directrix is [tex]y=-\frac 12[/tex]