Problem 342 - Calculus Volume 3

Question

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)

Step-by-Step Solution

Answer

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

Step 1: Identify the Center

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 these foci. To find the midpoint, use the midpoint formula: \(M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substituting the given coordinates, we calculate: \(\left(\frac{-7 + 1}{2}, \frac{2 + 2}{2}\right) = (-3, 2)\). Thus, the center of the ellipse is \((-3, 2)\).

Step 2: Determine the Length of the Focal Distance

The length of the focal distance \(2c\) is the distance between the two foci, which are given as \((-7, 2)\) and \((1, 2)\). To find the distance, use the distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Plugging in the values gives: \(d = \sqrt{(1 - (-7))^2 + (2 - 2)^2} = \sqrt{(8)^2 + 0} = 8\). Thus, \(2c = 8\), giving \(c = 4\).

Step 3: Calculate the Semi-Major and Semi-Minor Axes

The length of the major axis is given as 10, which is \(2a\), so \(a = \frac{10}{2} = 5\). The relationship between \(a\), \(b\) (semi-minor axis), and \(c\) (focal distance) in an ellipse is \(c^2 = a^2 - b^2\). Substituting \(a = 5\) and \(c = 4\), we have: \(4^2 = 5^2 - b^2\). Simplifying gives: \(16 = 25 - b^2\) resulting in \(b^2 = 9\), so \(b = 3\).

Step 4: Write the Cartesian Equation of the Ellipse

The standard form of the equation of an ellipse with the center at \( (h, k) \) is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Here, the center is \((-3, 2)\), \(a = 5\), and \(b = 3\). Plug these values into the equation: \(\frac{(x + 3)^2}{5^2} + \frac{(y - 2)^2}{3^2} = 1\) or \(\frac{(x + 3)^2}{25} + \frac{(y - 2)^2}{9} = 1\).

Key Concepts

Ellipse

An ellipse is a fascinating shape that looks like an elongated circle, and it's quite common in geometry. Unlike a circle where every point is equidistant from the center, an ellipse has two foci. The sum of the distances from any point on the ellipse to these two foci is always constant. This unique property gives it its particular shape and mathematical significance.

Ellipses are characterized by their major and minor axes:

• The major axis is the longest line that runs through the center and touches the widest part of the ellipse.
• The minor axis runs through the center but is perpendicular to the major axis and touches the narrowest part of the ellipse.

Understanding these properties helps in deriving the Cartesian equation of an ellipse, which allows us to identify its shape and position on a coordinate plane.

Midpoint Formula

To find the center of an ellipse, we can use the midpoint formula. The midpoint of a segment defined by two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using:

• \(M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)

In the exercise, the foci of the ellipse are \( (-7, 2) \) and \( (1, 2) \). By substituting these values into the formula:

• \( \left(\frac{-7 + 1}{2}, \frac{2 + 2}{2}\right) = (-3, 2) \)

The center of the ellipse is thus located at \( (-3, 2) \). Finding the center is crucial as it anchors the other geometric properties like the axes.

Distance Formula

The distance formula helps us determine the distance between two points in a plane. For points \( (x_1, y_1) \) and \( (x_2, y_2) \), it is expressed as:

• \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)

In our problem, the foci \( (-7, 2) \) and \( (1, 2) \) define the distance, giving us the focal distance. Using the formula:

• \( d = \sqrt{(1 - (-7))^2 + (2 - 2)^2} = \sqrt{(8)^2} = 8 \)

This distance is essential as it helps find the values of semi-major and semi-minor axes, fundamental in writing the ellipse's equation.

Semi-major and Semi-minor Axes

In an ellipse, the semi-major and semi-minor axes are vital in shaping the curve. They determine the length of the major and minor axes:

• \( a \) is the semi-major axis (half of the major axis).
• \( b \) is the semi-minor axis (half of the minor axis).

Given that the major axis is 10, \( a = \frac{10}{2} = 5 \). The relationship between \( a \), \( b \), and \( c \) (half the distance between the foci) is:\[c^2 = a^2 - b^2\]For our ellipse, \( c = 4 \), and substituting in the equation:

• \( 16 = 25 - b^2 \)
• \( b^2 = 9 \), therefore \( b = 3 \)

Calculating these axes provides us with the tools to write the precise equation of the ellipse and visualize its expansion on the Cartesian plane.