Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 11

Chapter 8, Problem 11

Graph each ellipse and locate the foci. x² = 1 – 4y²

Verified step by step guidance

Rewrite the given equation \(x^2 = 1 - 4y^2\) to the standard form of an ellipse equation. Start by moving all terms to one side: \(x^2 + 4y^2 = 1\).

Divide both sides of the equation by 1 to express it in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, it becomes \(\frac{x^2}{1} + \frac{y^2}{\frac{1}{4}} = 1\).

Identify the values of \(a^2\) and \(b^2\). From the equation, \(a^2 = 1\) and \(b^2 = \frac{1}{4}\). Since \(a^2 > b^2\), the major axis is along the x-axis.

Calculate the focal distance \(c\) using the relationship \(c^2 = a^2 - b^2\). Substitute the values to get \(c^2 = 1 - \frac{1}{4}\).

Locate the foci at points \((\pm c, 0)\) on the x-axis. These points represent the foci of the ellipse. Finally, sketch the ellipse using the intercepts and foci.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of an Ellipse

An ellipse is typically expressed in the form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h,k)\) is the center, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes. Understanding how to rewrite the given equation into this form is essential for graphing the ellipse.

Identifying the Orientation of the Ellipse

The ellipse can be oriented horizontally or vertically depending on whether the \(x^2\) or \(y^2\) term is associated with the larger denominator. Recognizing the orientation helps in correctly plotting the ellipse and locating its axes.

Locating the Foci of an Ellipse

The foci are two fixed points inside the ellipse, located along the major axis, found using \( c^2 = |a^2 - b^2| \). Knowing how to calculate \(c\) and place the foci relative to the center is crucial for completing the graph accurately.

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Foci and Vertices of an Ellipse

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