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Chapter 9: Problem 47

Determine the equation in standard form of the ellipse centered at the origin that satisfies the given conditions. Minor axis of length \(6 ;\) major axis of length \(14 ;\) major axis horizontal

Short Answer

Expert verified
The standard form equation for the given ellipse is \(\frac{x^2}{49} + \frac{y^2}{9} = 1\).

Step by step solution

01

Determine the Lengths of the Semi-Major and Semi-Minor Axes

The lengths of the major and minor axes are given, and both are double the lengths of the corresponding semi-axes (since an axis extends on both sides of the center). So, the semi-major axis length \(a\) is \(\frac{14}{2} = 7\), and the semi-minor axis length \(b\) is \(\frac{6}{2} = 3\).
02

Identify the Orientation of the Major Axis

The major axis is horizontal, indicating that the semi-major axis length \(a\) will be associated with the \(x\)-term in the equation of the ellipse.
03

Write the Standard Form of an Ellipse

The general form of an ellipse's equation centered at the origin (with \(h = 0\) and \(k = 0\)) is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). Here, substitute \(a = 7\) and \(b = 3\) into the equation.
04

Simplify the Equation of the Ellipse

Substitute \(a = 7\) and \(b = 3\) into the ellipse equation to give \(\frac{x^2}{7^2} + \frac{y^2}{3^2} = 1\), or simplifying further to \( \frac{x^2}{49} + \frac{y^2}{9} = 1\)

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