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

검증된 단계별 안내

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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주요 개념

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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.

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.

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.

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