Find the Properties x^2=-20y

Problem

x^2=−20*y

Solution

1. Identify the form of the parabola. The equation is in the form x^2=4*p*y which represents a vertical parabola opening either up or down.

2. Determine the value of p By comparing x^2=−20*y to x^2=4*p*y we set 4*p=−20

3. Solve for p Dividing both sides by 4 gives p=−5 Since p is negative, the parabola opens downward.

4. Find the vertex. For the standard form x^2=4*p*y the vertex is at the origin (0,0)

5. Locate the focus. The focus for a vertical parabola is (0,p) Substituting p=−5 the focus is (0,−5)

6. Determine the directrix. The directrix is the horizontal line y=−p Substituting p=−5 the equation is y=−(−5) which simplifies to y=5

7. Identify the axis of symmetry. Since the parabola opens vertically and the vertex is at the origin, the axis of symmetry is the yaxis, or x=0

8. Calculate the focal diameter (length of the latus rectum). The length is given by |4*p| Substituting p=−5 we get |−20|=20

Final Answer

Vertex: *(0,0), Focus: *(0,−5), Directrix: *y=5, Axis of Symmetry: *x=0, Focal Diameter: *20

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