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Chapter 9: Problem 78

Explain how to use \(y^{2}=8 x\) to find the parabola's focus and directrix.

Short Answer

Expert verified
The focus of the parabola is at the point (2,0) and the equation of the directrix is \(x = -2\).

Step by step solution

01

Identify the 'a' value

From the equation \(y^2=8x\) we observe that it is in the form \(y^2=4ax\). Comparing these two equations, we can deduce that 4a = 8, which leads us to find that \(a=2\)
02

Find the Focus

The focus of a parabola \(y^2 = 4ax\) is at the point (a, 0). Using the 'a' value we found in step 1, the focus coordinates are (2,0)
03

Find the Directrix

The equation of the directrix for a parabola in the form \(y^2=4ax\) is \(x = -a\). Using the 'a' value that we found in step 1, the directrix equation is \(x = -2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Focus of a Parabola
The focus of a parabola is a specific point that is crucial in understanding the parabola's geometry. For parabolas that open to the right or left, as in our equation \( y^2 = 8x \), the focus lies on the x-axis. The key role of the focus is that every point on the parabola is equidistant from the focus and the directrix, another essential component of parabola structure. To find the focus:
  • Recognize the parabola equation format. Here, it's \( y^2 = 4ax \).
  • Compare it to your equation and identify 'a'. In this case, \( 4a = 8 \), providing \( a = 2 \).
  • Place the focus at the point \( (a, 0) \). Thus, the focus for the equation \( y^2 = 8x \) is at \( (2, 0) \).
Understanding the focus provides insight into the shape and orientation of the parabola. When graphing, it can help to check your accuracy.
Directrix of a Parabola
The directrix of a parabola is an imaginary line that plays an essential role in defining its shape. Similar to the focus, every point on the parabola is equidistant from the directrix and the focus. For equations like \( y^2 = 8x \), which represent parabolas opening sideways:
  • The directrix line is vertical, parallel to the y-axis.
  • To find the directrix, use the value of 'a' identified earlier. The directrix formula is \( x = -a \).
  • With \( a = 2 \), the directrix of the parabola is \( x = -2 \).
Having both the directrix and focus allows you to understand the balanced nature of the parabola鈥攁 concept useful in many applications of geometry and physics.
Parabola Equation
A parabola equation is a mathematical representation used to graph these U-shaped curves. Parabolas can appear in different orientations based on their equations. For instance:
  • When the parabola's equation is in the form \( y^2 = 4ax \) or \( x^2 = 4ay \), it opens horizontally.
  • Look for values of 'a' to determine its precise shape and specific points like the focus and directrix.
  • For the equation \( y^2 = 8x \), identifying \( a = 2 \) allows us to not only place the focus and directrix but also aids in graphing.
Understanding the basic form and components of a parabola equation is vital in fields ranging from algebra to engineering, where parabolas are a central concept.

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Most popular questions from this chapter

In Exercises \(21-40\), eliminate the parameter \(t\). Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \(t .\) (If an interval for \(t\) is not specified, assume that \(-\infty < t < \infty .)\) $$x=2 \cos t, y=3 \sin t ; 0 \leq t<2 \pi$$

Exercises \(97-99\) will help you prepare for the material covered in the next section. Rewrite \(r=\frac{4}{2+\cos \theta}\) by dividing the numerator and the denominator by 2.

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Use the polar mode of a graphing utility with angle measure in radians . Unless otherwise indicated, use \(\theta \min =0, \theta \max =2 \pi,\) and \(\theta\) step \(=\frac{\pi}{48} .\) If you are not satisfied with the quality of the graph, experiment with smaller values for \(\theta\) step. Identify the conic that each polar equation represents. Then use a graphing utility to graph the equation. $$r=\frac{16}{4-3 \cos \theta}$$
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