53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²

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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.

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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.

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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.

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Convert 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.

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Parabolas as Conic Sections

Pratica correlata

Domanda del libro di testo

A polar conic section Consider the equation r² = sec2θ

a. Convert the equation to Cartesian coordinates and identify the curve.

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Domanda del libro di testo

40–41. {Use of Tech} Slopes of tangent lines

b. Find the slope of the lines tangent to the curve at the origin (when relevant).