Question

For the following exercises, find the Cartesian equation describing the given shapes. An ellipse with a major axis length of 10 and foci at \((-7,2)\) and \((1,2)\)

Short answer

The equation is \( \frac{(x+3)^2}{25} + \frac{(y-2)^2}{9} = 1 \).

Step-by-step solution

Identify the Center of the Ellipse

The foci of the ellipse are given as \((-7, 2)\) and \(1, 2)\). The center of the ellipse is the midpoint of the line segment joining the foci. Calculate the midpoint using the formula \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). The center is \((-3, 2)\).

Determine the Orientation of the Major Axis

Since the y-coordinates of both foci are the same ( 2 ), the major axis is horizontal.

Calculate the Half-Length of the Major Axis

The major axis has a given length of 10. Therefore, the semi-major axis (denoted as \('a'\)) is half of that: \('a = 5'\).

Calculate the Distance Between Foci (2c)

The distance between the foci, denoted as \(2c\), is \(1 - (-7) = 8\). Therefore, \(c = \frac{8}{2} = 4\).

Calculate the Semi-Minor Axis (b)

Use the relationship \(c^2 = a^2 - b^2\) to find the semi-minor axis. Plug in the values: \(4^2 = 5^2 - b^2\), leading to \(16 = 25 - b^2\) and \(b^2 = 9\). Thus, \(b = \sqrt{9} = 3\).

Write the Cartesian Equation of the Ellipse

The standard equation for a horizontally oriented ellipse is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \(h\) and \(k\) are the coordinates of the center. Substitute \(h = -3\), \(k = 2\), \(a = 5\), \(b = 3\): \(\frac{(x+3)^2}{25} + \frac{(y-2)^2}{9} = 1\).

Key concepts

Cartesian Coordinates

Cartesian coordinates are a fundamental concept used to pinpoint any location on a plane. This system uses two numbers to define positions. These numbers are known as the x-coordinate and y-coordinate. It’s akin to giving an exact address which tells you where a point is located on a grid.

The Cartesian coordinate system is arranged with two perpendicular lines called the x-axis and y-axis. Both axes intersect at the origin point \((0,0)\). The horizontal line is the x-axis, and the vertical line is the y-axis. Each point on the plane is represented by an ordered pair \((x, y)\).

When dealing with the Cartesian equation of an ellipse, the coordinates of its center, foci, and vertices are all defined using this system. In the example of the ellipse with foci at \((-7, 2)\) and \((1, 2)\), you can easily locate these points on a plane using their Cartesian coordinates, making geometry problems much more intuitive.

Ellipse Geometry

Ellipse geometry involves understanding the precise shape and properties of an ellipse. An ellipse looks like an elongated circle, and each section of it can be defined by specific geometric properties.

Key attributes of an ellipse include:

• **Major and Minor Axes**: The longest diameter of an ellipse is the major axis, and the shortest diameter is the minor axis.
• **Center**: This is the midpoint of both the major and minor axes.
• **Foci**: These are two distinct points inside the ellipse. The total distance from one focus to any point on the ellipse and back to the other focus is constant.

In our exercise, with a major axis length of 10, we find it is horizontal since the y-coordinates of the foci are the same. The semi-major axis is half the major axis, which is 5. You can also identify the semi-minor axis, calculated using the distances and properties of foci. Understanding these elements can help you visualize the ellipse shape accurately.

Conic Sections

Conic sections are shapes formed by cutting a cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas. Understanding conic sections allows you to explore a variety of geometric properties and equations.

An ellipse is one type of conic section. Specifically, it appears when a plane intersects a cone at an angle, resulting in a closed curve. The standard form of an ellipse’s equation helps represent its geometric properties succinctly.

• **Equation of an Ellipse**: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
• **Properties**: The parameters of the equation relate to its foci, axes, and center, defining the exact shape.

Knowing how to identify and use these characteristics is critical when working with ellipse geometry and its equation. For example, in our task, substituting known values into the ellipse equation provides an accurate mathematical representation of the ellipse.