Find the Properties x^2=8y

Problem

x^2=8*y

Solution

1. Identify the form of the parabola. The equation x^2=8*y is in the standard form x^2=4*p*y which represents a parabola that opens upward with its vertex at the origin.

2. Determine the value of p By comparing x^2=8*y to x^2=4*p*y we set 4*p=8

4*p=8

p=2

3. Locate the vertex. Since the equation is in the form x^2=4*p*y the vertex is at the origin.

Vertex=(0,0)

4. Find the focus. For a parabola opening upward, the focus is located at (0,p)

Focus=(0,2)

5. Determine the directrix. The directrix is a horizontal line given by the equation y=−p

Directrix:y=−2

6. Identify the axis of symmetry. Since the x term is squared, the parabola is symmetric about the yaxis.

Axis of Symmetry:x=0

7. Calculate the length of the latus rectum. The length is given by |4*p|

Latus Rectum Length=8

Final Answer

Vertex: *(0,0), Focus: *(0,2), Directrix: *y=−2, Axis of Symmetry: *x=0

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