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Chapter 6: Problem 15

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (卤5,0)\(;\) major axis of length 14

Short Answer

Expert verified
The standard form of the equation of the ellipse is \((x^2 / 49) + (y^2 / (576/49)) = 1\).

Step by step solution

01

Determine the Eccentricity

The distance from the center of an ellipse to its foci is represented by the letter \(c\). In this case, \(c = 5\). The length of the major axis is equal to \(2a\), where \(a\) is the semimajor axis. Therefore, we can determine that \(a = 14 / 2 = 7\). The eccentricity (e) of an ellipse is defined as \(c / a\), hence \(e = 5 / 7\).
02

Calculate the Semiminor Axis

To find the semiminor axis (b), we need to use the formula \(b = a \sqrt{1 - e^2}\). After substituting \(a = 7\) and \( e = 5 / 7 \), calculate \(b = 7 \sqrt{1 - (5/7)^2}\), which results in \(b = 24 / 7\).
03

Write the Standard Form of the Equation

Now that we have the values of the semimajor (a) and semiminor (b) axes, they can be substituted into the standard form of an ellipse equation centered at the origin: \((x^2 / a^2) + (y^2 / b^2) = 1\). Replacing \(a = 7\) and \(b = 24/7\), the equation becomes \((x^2 / 49) + (y^2 / (576/49)) = 1\).

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Most popular questions from this chapter

Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) \(y=a(x-h)^{2}+k, \quad a \neq 0\) (b) \((x-h)^{2}=4 p(y-k), \quad p \neq 0\) (c) \((y-k)^{2}=4 p(x-h), \quad p \neq 0\)

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(\frac{x^{2}}{5}+\frac{y^{2}}{9}=1\)

Use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for \(y\) and obtain two equations.) \(36 x^{2}+9 y^{2}+48 x-36 y-72=0\)

Show that \(a^{2}=b^{2}+c^{2}\) for the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) where \(a>0, b>0,\) and the distance from the center of the ellipse (0,0) to a focus is \(c\).

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(9 x^{2}+4 y^{2}-54 x+40 y+37=0\)=
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