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Chapter 6: Problem 57

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (0,2)\(;\) directrix: \(y=4\)

Short Answer

Expert verified
The standard form of the equation of the parabola with vertex at (0, 2) and directrix at y = 4 is \(y = 2 - \frac{x^2}{8}\).

Step by step solution

01

Find the distance a

The vertex of the parabola is at (0,2) and the directrix is at y=4. The distance a from the vertex to the directrix can be calculated as follows: a = k - y, where k is the y-coordinate of the vertex and y is the y-coordinate of the directrix. So in this case, a = 2 - 4 = -2.
02

Substitute h, k and a into the standard equation

The standard form of the equation of a parabola is \((x-h)^2 = 4a(y-k)\), where (h,k) is the vertex and a is the distance from the vertex to the directrix. Substituting the values h = 0, k = 2, and a = -2, the equation becomes \((x-0)^2 = 4(-2)(y-2)\). Simplifying this, we get \(x^2 = -8(y-2)\).
03

Rearrange the equation in standard form

The standard form of the equation is generally expressed with y as the subject. We can rearrange the equation from the previous step to be in that form: \(x^2 = -8(y-2)\) can be rewritten as \(y = 2 - \frac{x^2}{8}\), which is the standard form of the equation of the parabola.

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