Find the Properties x^2=-12y

Problem

x2=−12*y

Solution

1. Identify the form of the parabola. The equation x2=−12*y is in the standard form x2=4*p*y which represents a parabola that opens vertically.

2. Determine the value of p Set 4*p equal to the coefficient of y

4*p=−12

p=−3

3. Find the vertex. Since there are no h or k offsets in the equation, the vertex is at the origin.

Vertex=(0,0)

4. Determine the direction of opening. Since p<0 and the equation is x2=4*p*y the parabola opens downward.

5. Calculate the focus. For a vertical parabola with vertex (0,0) the focus is (0,p)

Focus=(0,−3)

6. Identify the directrix. The directrix is a horizontal line given by y=−p

y=−(−3)

y=3

7. Find the axis of symmetry. The axis of symmetry passes through the vertex and focus, which is the yaxis.

x=0

8. Determine the length of the latus rectum. The length is given by |4*p|

Length=|−12|=12

Final Answer

Vertex: *(0,0), Focus: *(0,−3), Directrix: *y=3, Axis of Symmetry: *x=0

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