53–57. Conic sections
c. Find the eccentricity of the curve.
y² - 4x² = 16

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Rewrite the given equation \(y^{2} - 4x^{2} = 16\) in the standard form of a conic section by dividing both sides by 16, resulting in \(\frac{y^{2}}{16} - \frac{4x^{2}}{16} = 1\), which simplifies to \(\frac{y^{2}}{16} - \frac{x^{2}}{4} = 1\).

Recognize that the equation is in the form \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\), which represents a hyperbola centered at the origin with the transverse axis along the y-axis.

Identify the values of \(a^{2}\) and \(b^{2}\) from the equation: \(a^{2} = 16\) and \(b^{2} = 4\).

Recall the formula for the eccentricity \(e\) of a hyperbola: \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus, and \(c^{2} = a^{2} + b^{2}\).

Calculate \(c\) using \(c^{2} = a^{2} + b^{2}\), then find the eccentricity \(e = \frac{c}{a}\). This will give the eccentricity of the hyperbola.

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Key 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 ellipses, parabolas, or hyperbolas. Each conic has a standard equation form; for example, hyperbolas often appear as x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1. Recognizing the type of conic from its equation is essential for further analysis.

Eccentricity of a Conic Section

Eccentricity (e) measures how much a conic deviates from being circular. For ellipses, 0 < e < 1; for parabolas, e = 1; and for hyperbolas, e > 1. It is defined as the ratio of the distance from a focus to a point on the curve over the perpendicular distance from that point to the directrix, and it characterizes the shape of the conic.

Rewriting and Identifying Parameters from the Given Equation

To find the eccentricity, the given equation y² - 4x² = 16 must be rewritten in standard form by dividing both sides by 16, yielding (y²/16) - (x²/4) = 1. This identifies a hyperbola with a² = 16 and b² = 4, allowing calculation of eccentricity using e = √(1 + b²/a²). Understanding this manipulation is key to solving the problem.

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