Problem 11 - College Algebra

Question

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. $$ \frac{x^{2}}{4}+\frac{y^{2}}{49}=1 $$

Step-by-Step Solution

Answer

Ellipse: major axis endpoints (0,7) and (0,-7), minor axis endpoints (2,0) and (-2,0), foci (0,3√5) and (0,-3√5).

Step 1: Identifying the Standard Form

The given equation is \( \frac{x^2}{4} + \frac{y^2}{49} = 1 \). This is the standard form of an ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a^2 = 4 \) and \( b^2 = 49 \).

Step 2: Finding the Lengths of Axes

To find \( a \), take the square root of \( a^2 \): \( a = \sqrt{4} = 2 \). Similarly, to find \( b \), take the square root of \( b^2 \): \( b = \sqrt{49} = 7 \). Since \( b > a \), the major axis is vertical and length of the major axis is \( 2b = 14 \), and the minor axis is horizontal with a length of \( 2a = 4 \).

Step 3: Determining the Endpoints of Axes

Since the ellipse is centered at the origin (\((0, 0)\)), the endpoints of the major axis are \((0, b) \) and \((0, -b)\), which are \((0, 7)\) and \((0, -7)\). The endpoints of the minor axis are \((a, 0)\) and \((-a, 0)\), which are \((2, 0)\) and \((-2, 0)\).

Step 4: Finding the Foci

The foci of the ellipse are along the major axis. Use the formula \( c^2 = b^2 - a^2 \) to find \( c \): \( c^2 = 49 - 4 = 45 \), so \( c = \sqrt{45} = 3\sqrt{5} \). The foci are at \((0, c)\) and \((0, -c)\), which are \((0, 3\sqrt{5})\) and \((0, -3\sqrt{5})\).

Key Concepts

Standard Form

The standard form of an ellipse equation is a simple way to recognize the shape and size of an ellipse in mathematical terms. When we see an equation like \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we know that it follows the structure of an ellipse's standard equation.
\( a^2 \) and \( b^2 \) are crucial values that dictate the ellipse's dimensions. A handy tip is to remember that the larger number between \( a^2 \) and \( b^2 \) designates the direction of the major axis. For instance, if \( a^2 = 4 \) and \( b^2 = 49 \), as in the equation \( \frac{x^2}{4} + \frac{y^2}{49} = 1 \), then the major axis is vertical because \( b^2 \) is greater.
Recognizing an ellipse from its equation helps you identify its characteristics and solve problems related to its shape, size, and position easily.

Major Axis

The major axis of an ellipse is the longer diameter that passes through its widest points. In terms of the equation, this axis is determined by the larger of the two values \( a^2 \) or \( b^2 \). In our specific case, since \( b^2 = 49 \) is greater than \( a^2 = 4 \), the major axis is vertical.

• Length of the major axis: calculated as \( 2b = 2 \times 7 = 14 \).
• Endpoints: located at the points \((0, 7)\) and \((0, -7)\) because the ellipse is centered at the origin.

These endpoints provide the farthest reach of the ellipse along this axis and help in visualizing its complete extent in the coordinate plane.

Minor Axis

The minor axis is the shorter diameter of the ellipse, crossing the ellipse at its narrow stretch. In the case of our standard ellipse equation \( \frac{x^2}{4} + \frac{y^2}{49} = 1 \), it is determined by the smaller value of \( a^2 \) compared to \( b^2 \), which means the minor axis is horizontal.

• Length of the minor axis: given by \( 2a = 2 \times 2 = 4 \).
• Endpoints: found at the coordinates \((2, 0)\) and \((-2, 0)\), assuming the ellipse is centered at the origin.

These pairs of points mark the narrowest distance of the ellipse, making it easier to draw or imagine the full outline of the shape.

Foci

An ellipse has two foci (singular: focus) which are crucial in defining the shape's curve. The position of the foci is always along the major axis. For the equation \( \frac{x^2}{4} + \frac{y^2}{49} = 1 \), the foci's distance from the center can be calculated using \( c^2 = b^2 - a^2 \).
Calculate \( c \) as follows:

• \( c^2 = 49 - 4 = 45 \)
• \( c = \sqrt{45} = 3\sqrt{5} \)

Thus, the foci are located at \((0, 3\sqrt{5})\) and \((0, -3\sqrt{5})\), indicating their vertical positioning along the major axis given the current origin-centered ellipse.
Knowing where the foci are helps in understanding the internal geometry of the ellipse and is pivotal for many real-world applications, such as satellite orbits and optical lenses.