Problem 16 - Precalculus

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Horizontal major axis; passes through the points (5,0) and (0,2)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \(\frac{x^2}{25} + \frac{y^2}{4}=1\).

Step 1: Determine the value of \(a\)

From the fact that the ellipse passes through the point (5,0) and has a major axis along the x-axis, we can say that \(a\) is equal to the x-coordinate of this point. So, \(a=5\).

Step 2: Determine the value of \(b\)

Since the ellipse also passes through the point (0,2) and the y-axis is the minor axis, we can say that \(b\) is equal to the y-coordinate of this point. So, \(b=2\).

Step 3: Write the standard form

Now that we know the values of \(a\) and \(b\), we can write the standard form of the equation of the ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2}=1 \). Substituting \(a=5\) and \(b=2\) into the equation, we get: \(\frac{x^2}{25} + \frac{y^2}{4}=1\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation is pivotal for identifying its geometric properties and understanding its shape. When an ellipse is centered at the origin, the standard form is expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. This equation can easily be related to the more familiar equation of a circle, \(x^2 + y^2 = r^2\), emphasizing that an ellipse can be viewed as a stretched circle.

In this context, determining the values of \(a\) and \(b\) allows us to precisely define the shape of the ellipse. Knowing that the ellipse passes through specific points provides us with enough information to calculate these parameters. Once we establish \(a\) and \(b\), the standard form becomes a simple substitution away, enabling us to visualize and analyze the ellipse in detail. The origin-centered ellipse passing through the given coordinates led us to the standard form equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), representing an ellipse with a stretched horizontal axis.

Major and Minor Axes

In every ellipse, the major and minor axes are two crucial perpendicular diameters that define its width and height. The major axis is the longest diameter of the ellipse and runs through its widest part, while the minor axis is the shorter one and crosses the narrower section. If an ellipse is oriented such that its major axis is horizontal, as in the exercise, this axis would parallel the x-axis and the minor axis would parallel the y-axis.

The lengths of these axes are defined by the values of \(a\) and \(b\) found in the standard equation. Specifically, \(a\) corresponds to half of the length of the major axis, and \(b\) to half of the minor axis. These two axes intersect at the center of the ellipse, and their lengths determine the shape's overall proportions. By setting \(a=5\) and \(b=2\), we deduced that the major axis is 10 units long, and the minor axis is 4 units in length — revealing that our ellipse is horizontally oriented with a greater span along the x-axis.

Coordinates of Ellipse Points

Given its geometric properties, an ellipse contains an infinite number of points, each conforming to the standard equation. The coordinates of these points provide critical information about the ellipse's boundary and can be determined by fulfilling the standard equation. By inserting a value for \(x\) or \(y\) into the equation, we can solve for its counterpart, effectively finding points along the ellipse's edge.

For example, the exercise illustrates this by establishing that points (5,0) and (0,2) lie on the ellipse. They tell us the maximum extent to which the ellipse stretches along the x and y axes, respectively. These coordinates are particularly significant because they align with the axes' endpoints and are, therefore, directly related to \(a\) and \(b\). Essentially, in the given equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), one can plug in different \(x\) or \(y\) values to compute a series of corresponding \(y\) or \(x\) values. This method can map out a collection of points that, when connected, visualize the shape and size of the ellipsis.