Textbook Question
53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1
23
views
16. Parametric Equations & Polar Coordinates
Conic Sections
2:12 minutesProblem 12.R.54a
Textbook Question
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²
Verified step by step guidance1
Recall the general forms of conic sections:
- Parabola: one variable is squared, the other is to the first power (e.g., \(y^2 = 4ax\) or \(x^2 = 4ay\)).
- Ellipse: both variables are squared with the same sign and different coefficients (e.g., \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)).
- Hyperbola: both variables are squared but with opposite signs (e.g., \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)).
Look at the given equation: \(x = 16y^2\). Notice that \(x\) is to the first power and \(y\) is squared.
Rewrite the equation to isolate zero on one side: \(x - 16y^2 = 0\). This shows only one variable is squared and the other is linear.
Since only one variable is squared and the other is linear, this matches the form of a parabola, where the squared term is on one side and the linear term on the other.
Therefore, conclude that the equation \(x = 16y^2\) represents a parabola.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Conic Sections and Their Standard Forms
Conic sections are curves obtained by intersecting a plane with a double-napped cone, resulting in parabolas, ellipses, or hyperbolas. Each conic has a standard equation form, such as y² = 4ax for parabolas, (x²/a²) + (y²/b²) = 1 for ellipses, and (x²/a²) - (y²/b²) = 1 for hyperbolas. Recognizing these forms helps classify the given equation.
Recommended video:
5:33
Parabolas as Conic Sections
Identifying a Parabola from Its Equation
A parabola is characterized by an equation where one variable is squared and the other is to the first power, typically in the form x = ay² or y = ax². The given equation x = 16y² fits this pattern, indicating it is a parabola opening along the x-axis. Understanding this helps distinguish it from ellipses and hyperbolas.
Recommended video:
3:03Convert Equations from Polar to Rectangular Example 2
Graphical Interpretation of Conic Sections
Graphing the equation provides visual insight into the conic's shape and orientation. For x = 16y², the parabola opens rightward since x depends on y² with a positive coefficient. Recognizing the direction and shape of the graph aids in confirming the type of conic section represented by the equation.
Recommended video:
5:33Parabolas as Conic Sections
Related Videos
Related Practice