Problem 41 - Precalculus: Mathematics for Calculus

Question

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Foci: \(F(\pm 1,0),\) vertices: \((\pm 2,0)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \).

Step 1: Identify the Type of Ellipse

Since both the foci and the vertices have their coordinates along the x-axis, the ellipse is horizontal. The general equation for a horizontal ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a > b \).

Step 2: Determine the Value of a

The vertices of the ellipse are at \((\pm a, 0)\). Given the vertices are \((\pm 2,0)\), we have \( a = 2 \).

Step 3: Determine the Value of c

The foci of the ellipse are at \((\pm c, 0)\). Given the foci are \((\pm 1, 0)\), we have \( c = 1 \).

Step 4: Calculate b using the relationship \(c^2 = a^2 - b^2\)

Using the formula \( c^2 = a^2 - b^2 \), we substitute \( c = 1 \) and \( a = 2 \): \( 1^2 = 2^2 - b^2 \)\( 1 = 4 - b^2 \)Solving for \( b^2 \):\( b^2 = 3 \).So, \( b = \sqrt{3} \).

Step 5: Write the Equation of the Ellipse

Substitute \( a^2 = 4 \) and \( b^2 = 3 \) into the ellipse equation:\[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \].

Key Concepts

Foci

In an ellipse, the foci (plural of focus) are two fixed points that help define its shape. They are crucial in understanding how the ellipse stretches across space. The foci are always located along the major axis of the ellipse.For a horizontal ellipse, like in our exercise, both foci lie on the x-axis. Here, they're located at \((\pm 1, 0)\). This means they're centered around the origin, specifically 1 unit away from it on each side. The distance between the foci and the center (0,0) is denoted as \(c\). Therefore, in this scenario, \(c = 1\).

• The closer the foci are to each other relative to the vertices, the more circular (less elongated) the ellipse is.
• The farther apart they are, the more elongated the ellipse becomes.

Vertices

The vertices of an ellipse are the points where the ellipse is widest. For a horizontal ellipse, the vertices lie along the major axis which, in this context, is the x-axis.Given the vertices at \((\pm 2, 0)\), each vertex is 2 units away from the center, meaning \(a = 2\). The value of \(a\) represents the distance from the center to one of the vertices along the major axis.

• Vertices are essential in determining the size of the ellipse.
• The value of \(a\) is always greater than the value of \(b\) in horizontal ellipses, ensuring the major axis is along the x-axis.

Ellipse

An ellipse is a smooth, closed curve that appears like an elongated circle. It is defined based on the sum of distances from any point on the ellipse to the two foci, which is constant. This unique property differentiates it from other geometric shapes.The standard form of an ellipse centered at the origin is given by the equation:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]The parameters \(a\) and \(b\) are key in determining the ellipse's dimensions:

• \(a\) is the distance from the center to a vertex along the major axis.
• \(b\) is the distance from the center to the other axis, known as the minor axis.

The relationship \(c^2 = a^2 - b^2\) helps in finding the value of \(b\) once \(a\) and \(c\) are known.

Horizontal Ellipse

A horizontal ellipse is an ellipse whose major axis is aligned with the x-axis. In simpler terms, it stretches more from left to right than it does up and down. The general equation takes the form:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a > b\).In the exercise, the equation obtained from the given vertices and foci was \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), indicating a horizontal orientation. Here's why this matters:

• A horizontal ellipse ensures the widest part aligns with the x-axis, influenced by the positions of the vertices.
• Knowing whether the ellipse is horizontal or vertical can change the perspectives of the distances and dimensions involved.