Find the Properties 16y=x^2

Problem

16*y=x^2

Solution

1. Identify the type of conic section. Since the equation is in the form x^2=4*p*y it represents a parabola that opens vertically.

2. Rewrite the equation in standard form x^2=4*p*y to find the value of p

x^2=16*y

4*p=16

p=4

3. Determine the vertex. Since there are no h or k offsets in the equation (x−h)^2=4*p*(y−k) the vertex is at the origin.

Vertex=(0,0)

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

Focus=(0,0+4)

Focus=(0,4)

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

y=0−4

y=−4

6. Identify the axis of symmetry. Since the parabola opens vertically and the vertex is at the origin, the axis of symmetry is the y-axis.

x=0

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

Focal Diameter=|16|

Focal Diameter=16

Final Answer

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

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