Graph each ellipse and locate the foci. x2/16+y2/4 = 1

Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

문제 1

8장, 문제 1

Graph each ellipse and locate the foci. x²/16+y²/4 = 1

검증된 단계별 안내

Identify the standard form of the ellipse equation: \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Here, \(a^{2} = 16\) and \(b^{2} = 4\).

Determine which axis is the major axis by comparing \(a^{2}\) and \(b^{2}\). Since \(16 > 4\), the major axis is along the x-axis, and \(a = 4\), \(b = 2\).

Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values to get \(c^{2} = 16 - 4\).

Find the value of \(c\) by taking the square root: \(c = \sqrt{12}\). This gives the distance from the center to each focus along the major axis.

Locate the foci at points \((\pm c, 0)\) on the x-axis, and sketch the ellipse centered at the origin with vertices at \((\pm a, 0)\) and co-vertices at \((0, \pm b)\).

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주요 개념

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Standard Form of an Ellipse

An ellipse in standard form is written as (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Understanding this form helps identify the shape and orientation of the ellipse on the coordinate plane.

Major and Minor Axes

The major axis is the longest diameter of the ellipse, while the minor axis is the shortest. The values of a and b determine which axis is major or minor, affecting the ellipse's width and height and guiding accurate graphing.

Foci of an Ellipse

The foci are two fixed points inside the ellipse such that the sum of distances from any point on the ellipse to the foci is constant. Their positions are found using c² = a² - b², where c is the distance from the center to each focus along the major axis.