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Chapter 7: Problem 30

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: \((0,-2),(0,2) ; x\) -intercepts: \(-2\) and 2

Short Answer

Expert verified
The standard form equation for the ellipse is \(x^2 =4.\)

Step by step solution

01

Determine the distance between the foci

The foci are given as (0,-2) and (0,2). We subtract the y-coordinates of these points to find the distance between them, which gives us the length of the major axis (2c). So, we have \(2c = |2 - (-2)| = 4.\) Therefore, the distance from the center to each focus (c) is \(c = 2.\)
02

Determine the length of the minor axis

The x-intercepts are given as -2 and 2, so the length of the minor axis (2a) is the distance between these points. Therefore, \(2a = |2 - (-2)| = 4,\) which implies that \(a = 2.\)
03

Find the semi-major axis (b)

We know that the relationship between a, b, and c in an ellipse is given by \(c^2 = a^2 - b^2.\) We can use this equation to find the semi-major axis (b), by substituting the values of c and a we found in steps 1 and 2. So, we get: \(2^2 = 2^2 - b^2 \Rightarrow b^2 = 0.\) Therefore, \(b = 0.\)
04

Write the standard form of the ellipse

The standard form of the equation of an ellipse is \((x-h)^2/a^2 + (y-k)^2/b^2 = 1,\) where (h,k) is the center of the ellipse. Since the center of the ellipse is at the origin (0,0), the equation becomes \(x^2/a^2 + y^2/b^2 = 1.\) Substituting the values of a and b we found in Steps 2 and 3, the equation becomes \(x^2/2^2 + y^2/0 = 1,\) which simplifies to \(x^2/4 = 1\) or \(x^2 =4.\)

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Most popular questions from this chapter

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length \(8 ;\) length of minor axis \(=4 ;\) center: \((0,0)\)

Graph each ellipse and give the location of its foci. $$\frac{x^{2}}{25}+\frac{(y-2)^{2}}{36}=1$$

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length \(12 ;\) length of minor axis \(=6 ;\) center: \((0,0)\)

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(y-2)^{2}-(x+3)^{2}=5$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$7 x^{2}=35-5 y^{2}$$
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