Find the Properties x^2=20y

Problem

x2=20*y

Solution

1. Identify the type of conic section. Since the equation is in the form x2=4*p*y it represents a parabola that opens vertically.

2. Determine the value of p by comparing the given equation x2=20*y to the standard form x2=4*p*y

4*p=20

p=5

3. Locate the vertex. For the standard form x2=4*p*y the vertex is at the origin.

Vertex=(0,0)

4. Find the focus. Since the parabola opens upward (p>0, the focus is located at (0,p)

Focus=(0,5)

5. Determine the directrix. The directrix is a horizontal line located at y=−p

Directrix:y=−5

6. Identify the axis of symmetry. For a parabola opening vertically with a vertex at the origin, the axis of symmetry is the y-axis.

Axis of Symmetry:x=0

7. Calculate the length of the latus rectum using the formula |4*p|

Latus Rectum Length=|20|=20

Final Answer

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

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