Find the Properties x^2=-24y

Problem

x2=−24*y

Solution

1. Identify the form of the parabola. The equation is in the form x2=4*p*y which represents a vertical parabola opening either up or down.

2. Determine the value of p By comparing x2=−24*y to x2=4*p*y we set 4*p=−24

3. Solve for p

4*p=−24

p=−6

4. Find the vertex. Since the equation is in the standard form x2=4*p*y without any translations, the vertex is at the origin.

Vertex=(0,0)

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

6. Find the focus. For a vertical parabola, the focus is located at (0,p)

Focus=(0,−6)

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

y=−(−6)

y=6

8. Identify the axis of symmetry. The axis of symmetry passes through the vertex and focus, which is the y-axis.

x=0

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

|4*(−6)|=24

Final Answer

x2=−24*y⇒Vertex: *(0,0), Focus: *(0,−6), Directrix: *y=6, Axis of Symmetry: *x=0

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