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Chapter 10: Problem 90

Find the standard form of the equation of the parabola with the given characteristics. Focus: (0,0)\(;\) directrix: \(y=8\)

Short Answer

Expert verified
The standard form of the equation of the given parabola is \(y=-4x^2\).

Step by step solution

01

Identify Focus and Directrix

The given focus is at (0,0), and the equation of the directrix is \(y=8\).
02

Define the Vertex

The vertex of the parabola is halfway between the focus and the directrix. This makes the vertex also at (0,0), the same as the focus.
03

Find the value of 'p'

The value of 'p' in the standard equation of a parabola represents half the distance between the focus and directrix. Here, \(p=-4\), negative because the parabola opens downwards.
04

Write the Standard Form of the Parabola

Our parabola is vertical and has a vertex at the origin (0,0), so its equation according to the general formula \(y=a(x-h)^2 + k\) will be simplified to \(y=px^2\), substituting \(p=-4\), the equation of the parabola becomes \(y=-4x^2\).

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Most popular questions from this chapter

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &(20,0),(4, \pi)\end{array}$$

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{14}{14+17 \sin \theta}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-8 y=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &(5, \pi)\end{array}$$

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{6}{2+\sin \theta}$$
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