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Chapter 11: Problem 58

Find an equation for each ellipse. Graph the equation. Foci at (1,2) and (-3,2)\(; \quad\) vertex at (-4,2)

Short Answer

Expert verified
The equation is \(\frac{(x + 1)^2}{9} + \frac{(y - 2)^2}{5} = 1\).

Step by step solution

01

Determine the Center

The center of an ellipse is the midpoint between the foci. The foci are at (1,2) and (-3,2). Calculate the midpoint: Midpoint = \(\frac{(1 + (-3))}{2}, \frac{(2 + 2)}{2}\) = (-1,2)
02

Calculate the Distance Between the Center and a Focus

The distance between the center (-1,2) and one of the foci (1,2) is \(c\). Use the distance formula: \[c = \sqrt{(1 - (-1))^2 + (2 - 2)^2} = 2\]
03

Calculate the Distance Between the Center and the Vertex

The vertex is at (-4,2), and the center is at (-1,2). The distance between them is \(a\). Use the distance formula: \[a = \sqrt{(-4 - (-1))^2 + (2 - 2)^2} = 3\]
04

Find the Value of \(b\)

Use the relationship \(a^2 = b^2 + c^2\) to find \(b\). Plug in the values of \(a\) and \(c\): \[3^2 = b^2 + 2^2\] \[9 = b^2 + 4\] \[b^2 = 5\]
05

Write the Equation

For a horizontal ellipse centered at (-1,2), the equation is in the form: \[\frac{(x + 1)^2}{a^2} + \frac{(y - 2)^2}{b^2} = 1\]Substitute \(a^2 = 9\) and \(b^2 = 5\): \[\frac{(x + 1)^2}{9} + \frac{(y - 2)^2}{5} = 1\]
06

Graph the Ellipse

Use the center (-1,2), vertices at (-4,2) and (2,2), and co-vertices to plot the ellipse on a coordinate plane. Recognize that the length of the major axis is 2a=6 and the length of the minor axis is 2b鈮�4.47.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ellipse center
The center of an ellipse is a critical concept in understanding and plotting this geometric shape. The center represents the midpoint between the foci of the ellipse.
To find the center, you calculate the midpoint between the foci using the midpoint formula. For our ellipse, the foci are at (1, 2) and (-3, 2). So,

Midpoint = \(\frac{(1 + (-3))}{2}, \frac{(2 + 2)}{2}\) = (-1, 2).
This point (-1, 2) is the center of the ellipse.
Understanding this helps in sketching the ellipse correctly and in deriving its equation.
foci of ellipse
The foci (singular: focus) of an ellipse are two fixed points located along the major axis of the ellipse. The sum of the distances from any point on the ellipse to the foci is constant.

For our example:
  • Foci are at points (1, 2) and (-3, 2).
  • The center, as found earlier, is at (-1, 2).
To understand their importance, we calculate the distance from the center to a focus, known as 'c'.

Let's use the distance formula:
\[c = \sqrt{(1 - (-1))^2 + (2 - 2)^2} = 2\]
This value (c) helps us find other crucial parameters of the ellipse, such as the lengths of the major and minor axes.
ellipse vertices
Vertices are the points where the ellipse intersects the major axis. In other words, they are the endpoints of the longest diameter of the ellipse.

To find the vertices in our ellipse, use the vertex at (-4, 2) and the center (-1, 2). We use the distance formula to determine 'a', the distance from the center to a vertex:
\[a = \sqrt{(-4 - (-1))^2 + (2 - 2)^2} = 3\]
  • So, one vertex is at (-4, 2),
  • and the other vertex is at (2, 2).
These vertices help us define the major axis and will be part of the ellipse's equation.
distance formula
The distance formula is essential in finding key features of an ellipse. It helps in calculating the lengths between different points, like the center to foci, or center to vertices.
The formula is:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
For our example, calculate the distance between the foci and the center to find 'c':
\[c = \sqrt{(1 - (-1))^2 + (2 - 2)^2} = 2\]
Then, calculate the distance between the center and the vertex to find 'a':
\[a = \sqrt{(-4 - (-1))^2 + (2 - 2)^2} = 3\]
These distances are critical for understanding the ellipse's geometry and formulating its equation correctly.

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Most popular questions from this chapter

Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. \(y^{2}-9 x^{2}=9\)

Sean throws a baseball with an initial speed of 145 feet per second at an angle of \(20^{\circ}\) to the horizontal. The ball leaves Sean's hand at a height of 5 feet. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long is the ball in the air? (c) Determine the horizontal distance that the ball travels. (d) When is the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a).

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Nuclear Power Plant Some nuclear power plants utilize "natural draft" cooling towers in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling tower has a base diameter of 400 feet, and the diameter at its narrowest point, 360 feet above the ground, is 200 feet. If the diameter at the top of the tower is 300 feet, how tall is the tower?

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