Problem 13 - 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: \((\pm7, 0); \quad\) foci: \((\pm2, 0)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse with the given vertices and foci is \[ \frac{x^2}{49} + \frac{y^2}{45} = 1. \]

Step 1: Identify a and c

The vertices are \((\pm7, 0)\) and foci are \((\pm2, 0)\). This suggests that the ellipse is horizontal, and a can be determined as half the distance between the vertices, and c as half the distance between the foci. Therefore, the values are \(a=7\) and \(c=2\).

Step 2: Compute b

Now, determine b through using the Pythagorean relationship \(a^2 = b^2 + c^2\). Substituting the known values, the equation becomes \(7^2= b^2 + 2^2\), resulting in \(b^2 = 7^2 - 2^2 = 45\), thus \(b=\sqrt{45} = 3\sqrt{5}\).

Step 3: Assemble the equation

Now that a, b, and c have been found, these values can be put into the standard form for an ellipse. The equation of the ellipse is thus \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \] which can be written as \[ \frac{x^2}{7^2} + \frac{y^2}{(3\sqrt{5})^2} = 1. \] Simplifying this equation gives \[ \frac{x^2}{49} + \frac{y^2}{45} = 1. \]

Key Concepts

Standard Form of Ellipse

To fully understand the concept of an ellipse, it's important to get acquainted with its standard form equation. The standard form of an ellipse centered at the origin is expressed as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here:

• \(a\) is the semi-major axis length, which represents half the length of the longest diameter of the ellipse.
• \(b\) is the semi-minor axis length, that is half the length of the shortest diameter.

The axes of an ellipse can be either horizontal or vertical based on the relative sizes of \(a\) and \(b\). If \(a > b\), the ellipse is horizontal, and if \(a < b\), it is vertical. For the problem given, since the vertices align with the x-axis, the ellipse is clearly horizontal, with its equation taking the form:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] adjusted as \(\frac{x^2}{49} + \frac{y^2}{45} = 1\), showing a clear picture of its shape and orientation along the x-axis.

Vertices of Ellipse

The vertices of an ellipse are crucial points that define the extent of its major axis. They serve as the bounds of the ellipse along this axis. The vertices given are \((\pm7, 0)\). This indicates our major axis lies along the x-axis, with the vertices horizontally 7 units away from the center at the origin.- **Vertices**: - Right vertex: \((7, 0)\) - Left vertex: \((-7, 0)\)
For any ellipse centered at the origin, the vertices can be expressed as \((\pm a, 0)\) or \((0, \pm a)\), based on whether the ellipse is horizontal or vertical, respectively. In this case, since the major axis is horizontal, the vertices conform to \((\pm a, 0)\). Therefore, \(a = 7\).
This vertex placement not only confirms the major axis's direction but also influences how we identify the ellipse's orientation in its coordinate plane.

Foci of Ellipse

Foci are another unique feature of ellipses, crucial to understanding the geometric properties of this shape. The ellipse in question has foci located at \((\pm2, 0)\). This means the foci are positioned along the x-axis, indicating the ellipse is elongated horizontally.- **Foci**: - Right focus: \((2, 0)\) - Left focus: \((-2, 0)\)
In mathematical terms, foci of an ellipse are defined such that the sum of the distances from any point on the ellipse to the two foci remains constant. The relationship involving the foci is governed by the equation \(c^2 = a^2 - b^2\), where \(c\) is the distance from the center to each focus. With \(a = 7\) and \(c = 2\), this Pythagorean relationship helps in finding \(b\) that completes the equation:\[ b^2 = a^2 - c^2 = 7^2 - 2^2 = 49 - 4 = 45 \]Thus, \(b = \sqrt{45} = 3\sqrt{5}\).
Understanding the foci locations not only aids in proper equation formation but also provides insight into how ellipses behave relative to their focal points.