Find the Properties y^2=-6x

Problem

y^2=−6*x

Solution

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

2. Determine the direction of the opening. Because the coefficient of x is negative (−6, the parabola opens to the left.

3. Find the value of p by setting the coefficient equal to 4*p

4*p=−6

p=−3/2

4. Identify the vertex. Since there are no h or k offsets in the equation y^2=−6*x the vertex is at the origin.

Vertex=(0,0)

5. Calculate the focus. For a horizontal parabola, the focus is located at (p,0)

Focus=(−3/2,0)

6. Determine the equation of the directrix. The directrix is a vertical line located at x=−p

x=−(−3/2)

x=3/2

7. Identify the axis of symmetry. Since the parabola opens horizontally and the vertex is at the origin, the axis of symmetry is the x-axis.

y=0

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

|4*p|=|−6|=6

Final Answer

Vertex: *(0,0), Focus: *(−3/2,0), Directrix: *x=3/2, Axis of Symmetry: *y=0

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