Problem 14 - Precalculus

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (±2,0)\(;\) major axis of length 10

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \( \frac{x^2}{25} + \frac{y^2}{21} = 1 \)

Step 1: Determine Semi-Major axis 'a'

Given the length of the major axis is 10, we know that the major axis is basically 2 times 'a'. So, the semi-major axis 'a' is half the length of the major axis. So, \(a = 10/2 = 5\)

Step 2: Calculate 'c'

The given locations of the foci are at (±2,0). 'c' is the distance from the center to each focus, so here, \(c = 2\)

Step 3: Calculate Semi-Minor axis 'b'

We can find 'b' using the relationship between 'a', 'b', and 'c' in an ellipse: \(c^2 = a^2 - b^2\). Solving this for 'b' gives: \(b = \sqrt{a^2 - c^2} = \sqrt{5^2 - 2^2} = \sqrt{21}\)

Step 4: Write down the Standard Form of the Equation of the Ellipse

The standard form of the equation of an ellipse with center at origin, major axis along the x-axis is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]. Substituting 'a' and 'b': we get \[ \frac{x^2}{5^2} + \frac{y^2}{21} = 1 \]

Key Concepts

Standard Form of an Ellipse

Understanding the standard form of an ellipse is crucial for solving problems related to this conic section. The standard form provides a clear representation of the ellipse's properties, including its axes lengths and orientation. An ellipse centered at the origin with the major axis aligned along the x-axis is represented by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

In the problem at hand, by applying the standard form, students can identify the values of \(a\) and \(b\) that satisfy the conditions given for the ellipse, leading to the equation \( \frac{x^2}{5^2} + \frac{y^2}{21} = 1 \) for the presented example. This form is efficient and simplifies the process of graphing the ellipse or calculating its area.

Semi-Major Axis

The semi-major axis of an ellipse plays a pivotal role, as it represents half the length of the longest diameter of the ellipse. The value of the semi-major axis, denoted by \(a\), also determines the size and shape of the ellipse. In the step by step solution, we find that for an ellipse with a major axis of length 10, the semi-major axis \(a\) would be half this length, equating to 5.

The semi-major axis is not only a measure of size but also impacts the eccentricity of the ellipse, which dictates how 'stretched' it appears. A larger \(a\) relative to \(b\) would result in a more elongated shape. Recognizing the role of the semi-major axis helps students to better visualize the geometry of ellipses.

Semi-Minor Axis

Similarly, the semi-minor axis, represented by \(b\), is half the length of the ellipse's shortest diameter. It stands perpendicular to the semi-major axis and together, they define the limits of the ellipse. To calculate \(b\), one uses the relationship \(c^2 = a^2 - b^2\), which connects both the axes with the distance to the foci, denoted by \(c\).

In the proposed exercise, after computing that \(c = 2\) using the foci given, the semi-minor axis is found to be \(\sqrt{a^2 - c^2} = \sqrt{5^2 - 2^2} = \sqrt{21}\). This value helps to fully describe the ellipse's dimension and is essential for graphing and analyzing the properties of the ellipse.

Foci of an Ellipse

The foci (plural of focus) of an ellipse are two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the foci is constant. This distinctive feature of ellipses helps distinguish them from other conic sections. The distance from the center of the ellipse to each focus, known as \(c\), is related to the lengths of the semi-major and semi-minor axes.

For the ellipse described in the exercise, the foci are given as (±2,0). Using the positions of the foci, one can deduce the value of \(c\), which is critical for calculating \(b\) when \(a\) is known. Here, \(c = 2\), which conforms to the general formula for an ellipse centered at the origin. The foci play a significant role in the ellipse's construction and are fundamental in applications ranging from astronomy to engineering.