Find the Properties x^2=16y

Problem

x2=16*y

Solution

1. Identify the form of the parabola. The equation is in the standard form x2=4*p*y which represents a parabola that opens upward with its vertex at the origin (0,0)

2. Determine the value of p by comparing the given equation x2=16*y to the standard form x2=4*p*y

4*p=16

p=4

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

Focus=(0,4)

4. Find the directrix. The directrix is a horizontal line given by the equation y=−p

Directrix:y=−4

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

Axis of Symmetry:x=0

6. Calculate the focal diameter (length of the latus rectum). This is given by the absolute value of 4*p

Focal Diameter=|16|=16

Final Answer

Vertex: *(0,0), Focus: *(0,4), Directrix: *y=−4, Axis of Symmetry: *x=0

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