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Chapter 10: Problem 18

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (卤2,0)\(;\) major axis of length 10

Short Answer

Expert verified
The equation of the ellipse is \(\frac{x^{2}}{25} + \frac{y^{2}}{21} = 1\).

Step by step solution

01

Finding the value of c and a

The foci of an ellipse are \((\pm c, 0)\), so given that the foci are \((\pm2,0)\), we know that \(c = 2\). Also, the length of the major axis is 10, which is twice the distance from the center to the ellipse, or \(2a\). Therefore, \(a = \frac{10}{2} = 5\).
02

Solve for \(b^{2}\)

The relationship between \(a, b, c\) for an ellipse is given by \(c^{2} = a^{2} - b^{2}\). Substituting our known values we get \(2^{2} = 5^{2} - b^{2}\). Solving this equation for \(b^{2}\), we find that \(b^{2} = 5^{2} - 2^{2} = 21\).
03

Formulating the equation

Now that we have the values for \(a^{2}\) and \(b^{2}\), we can substitute these into the general form of the ellipse equation, which is \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Hence the equation of the ellipse is \(\frac{x^{2}}{5^{2}} + \frac{y^{2}}{21} = 1\).

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Most popular questions from this chapter

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}-2 a x=0$$

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$\theta=\frac{7 \pi}{6}$$

Convert the polar equation to rectangular form. $$r=2 \cos \theta$$

Convert the polar equation \(r=2(h \cos \theta+k \sin \theta)\) to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{5}{1-\sin \theta}$$
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