What are the focus and directrix of the parabola defined by the equation x = 1 + 8y^2?

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To find the focus and directrix of the given parabola described by the equation x = 1 + 8y2, we start by rewriting the equation in the standard form of a parabola. The standard form for a parabola that opens to the right is:

x = a(y - k)2 + h

In this case, comparing with our equation:

  • h = 1
  • k = 0
  • a = 8

With these values, we can identify the vertex of the parabola:

  • Vertex (h, k): (1, 0)

Next, to find the focus and directrix, we can use the following formulas:

  • Focus: (h + p, k), where p is the distance from the vertex to the focus.
  • Directrix: x = h – p.

For a parabola in the form x = a(y – k)2 + h, p is calculated as:

p = 1/(4a)

Substituting our value of a: 8:

p = 1/(4 * 8) = 1/32

Now we can find the focus and the directrix:

  • Focus: (1 + 1/32, 0) = (1.03125, 0)
  • Directrix: x = 1 – 1/32 = 1 – 0.03125 = 0.96875

In summary:

  • Focus: (1.03125, 0)
  • Directrix: x = 0.96875

These points and lines help describe the geometric properties of the parabola’s shape.

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