Find the Properties x^2=24y

Problem

x^2=24*y

Solution

1. Identify the form of the parabola. The equation x^2=24*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=24*y to x^2=4*p*y we set 4*p=24

4*p=24

p=6

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=(0,6)

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

Directrix:y=−6

6. Identify the axis of symmetry. Since the x term is squared and the vertex is at the origin, the axis of symmetry is 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=|24|=24

Final Answer

Vertex: *(0,0), Focus: *(0,6), Directrix: *y=−6, Axis of Symmetry: *x=0

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