Find the Properties y^2=32x

Problem

y2=32*x

Solution

1. Identify the type of conic section. The equation is in the form y2=4*p*x which represents a parabola opening to the right with its vertex at the origin (0,0)

2. Determine the value of p by comparing the given equation y2=32*x to the standard form y2=4*p*x

4*p=32

p=8

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

Focus=(8,0)

4. Find the directrix. The directrix is a vertical line given by the equation x=−p

x=−8

5. Identify the axis of symmetry. Since the y term is squared and the vertex is at the origin, the axis of symmetry is the x-axis.

y=0

6. Calculate the length of the latus rectum using the formula |4*p|

Length=|32|=32

Final Answer

Vertex: *(0,0), Focus: *(8,0), Directrix: *x=−8, Axis of Symmetry: *y=0, Latus Rectum Length: *32

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