Problem 17 - Precalculus with Limits

Question

In Exercises 11-18, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: \((0, \pm5); \quad\) passes through the point \((4, 2)\)

Step-by-Step Solution

Answer

The standard form equation of the given ellipse is \[x^2/16 + y^2/25 = 1\].

Step 1: Extract Information from Vertices

The vertices are given as (0, ±5). In the standard equation of the ellipse, this corresponds to \(b\), as the longer axis (or the Vertices) in this case is vertical. So, \(b\) is 5.

Step 2: Use Given Point to Find 'a'

We know that the ellipse goes through the point (4,2). Substituting \(x = 4\) and \(y = 2\) in the standard equation, we get \(16/a^2 + 4/b^2 = 1\). We already found that \(b = 5\), so substituting \(b = 5\) in the above equation, we get \(16/a^2 + 4/25 = 1\). Solving this equation, we find \(a^2 = 16\). So, \(a = \pm4\). As 'a' represents a distance here, we'll take the positive root \(a = 4\).

Step 3: Write the Standard Form of the Equation

Now that we have the values of \(a\) and \(b\), we can substitute these into equation of ellipse. This gives the standard form equation as \(x^2/4^2 + y^2/5^2 = 1\) or \[x^2/16 + y^2/25 = 1\].

Key Concepts

Standard Form of an Ellipse

The equation of an ellipse can be expressed in its standard form, which simplifies understanding its properties. The equation for an ellipse with its center at the origin \( (0,0) \) is given by: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where:

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

The standard form allows us to easily see the lengths of the axes and how the ellipse is positioned in the coordinate plane. Additionally, the terms where the denominator is larger indicate the direction of the longer axis, either horizontal or vertical.

Vertices

Vertices are key points on an ellipse. They help define its shape and orientation. In the given problem, the vertices are at \( (0, \pm5) \). This arrangement indicates that the major axis, or the longer axis, is vertical.

• The distance from the center to a vertex along this axis equals \( b \) (the semi-major axis length).

For this ellipse, the value of \( b \) is 5, as found from the distance from the center to either vertex. The vertices are a crucial part of identifying how the ellipse is stretched.

Center at Origin

The center of an ellipse is the point from which the shape is symmetrically laid out on the coordinate plane. For the center at the origin, its coordinates are given as \( (0,0) \). This makes calculations straightforward, since the symmetry is easier to deal with in mathematical equations. Placing the center at the origin simplifies the ellipse's equation, as it results in: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]without needing any horizontal or vertical shifts in the equation.

Axes Lengths

Axes lengths in an ellipse tell us how much the shape is stretched horizontally and vertically. They are defined by parameters \( a \) and \( b \).

• \( a \) is the semi-major axis, which in this exercise is 4, indicating the width of the ellipse in the horizontal direction.
• \( b \) is the semi-minor axis, which is 5, creating the height of the ellipse vertically.

These values derive from the given vertices and the condition that the ellipse passes through a specific point. When you substitute these into the standard form equation, they define the exact shape and position of the ellipse.