Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ 4 x^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

Vertices: (0,4) and (0,-4). Foci: (0,2√3) and (0,-2√3).

Step 1: Write the Standard Form of the Ellipse

To convert the given ellipse equation \(4x^2 + y^2 = 16\) into its standard form, divide the entire equation by 16. That gives us:\[\frac{x^2}{4} + \frac{y^2}{16} = 1.\]

Step 2: Identify Parameters

The standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) allows us to identify the parameters: \(a^2 = 4\) and \(b^2 = 16\). Thus, \(a = 2\) and \(b = 4\). Here, \(b > a\), indicating a vertical major axis.

Step 3: Find the Vertices

The vertices of an ellipse with a vertical major axis are at \((h, k \pm b)\), where \(h\) and \(k\) are the center coordinates. Here, the center is at \((0, 0)\), so the vertices are \((0, \pm 4)\).

Step 4: Find the Foci

The distance \(c\) from the center to each focus is given by \(c^2 = b^2 - a^2\). Substituting the values, we find \(c^2 = 16 - 4 = 12\), so \(c = \sqrt{12} = 2\sqrt{3}\). Therefore, the foci are at \((0, \pm 2\sqrt{3})\).

Step 5: Sketch the Graph

To sketch the ellipse, plot the center at (0,0), the vertices at (0,4) and (0,-4), and the foci at (0,2√3) and (0,-2√3). Draw an ellipse around these points, ensuring it is taller than it is wide, reflecting the vertical orientation.

Key Concepts

Vertices of an Ellipse

For an ellipse, the vertices are special points that lie on the major axis, which is the longest diameter of the ellipse. The vertices represent the most extreme positions on the ellipse for given axes directions.
When an ellipse is centered at the origin (0, 0), and it has a vertical major axis, the standard form is given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( b > a \).
In this case, the vertices are located at \((0, b)\) and \((0, -b)\).

• The values \(a\) and \(b\) are derived from the denominators of the squared terms in the ellipse's equation.
• For the ellipse \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \), the vertices are at (0, 4) and (0, -4).

The vertices help in defining the shape and span of the ellipse along its major axis.

Foci of an Ellipse

The foci of an ellipse are two important points lying on the major axis inside the ellipse. They are used to define the unique shape an ellipse takes, based on the principle that the sum of the distances from any point on the ellipse to the foci is constant.
To determine the foci positions, we use the relationship \( c^2 = b^2 - a^2 \), where \( c \) is the distance from the center to each focus.
For the ellipse \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \):

• Calculating \( c \), we first find \( c^2 = 16 - 4 = 12 \).
• Thus, \( c = \sqrt{12} = 2\sqrt{3} \).
• The foci are located at \((0, 2\sqrt{3})\) and \((0, -2\sqrt{3})\).

The presence of foci is crucial for understanding how ellipses stretch and compress along the major axis.

Standard Form of an Ellipse

The standard form of an ellipse is a structured way of presenting its equation that allows us easily to identify its main attributes, such as the direction of the major axis and lengths of the semi-axes.
An ellipse can have a horizontal or vertical major axis. The general standard form is:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Here, if \( b > a \), the ellipse has a vertical major axis, and if \( a > b \), it has a horizontal major axis.
This form arises by dividing the original equation by the constant term to isolate 1 on the right side.
For the given equation \( 4x^2 + y^2 = 16 \):

• After dividing every term by 16, we get \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \).

This manipulation highlights the ellipse's parameters, making further calculations such as finding vertices and foci straightforward.

Graphing Ellipses

Graphing an ellipse effectively involves plotting its key components - center, vertices, and foci - and then sketching the curve around these points.
Begin by identifying the center, which, in the given standard form, is typically at the origin, \((0, 0)\).

• From the equation \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \), note: center (0,0), vertices (0, 4) and (0, -4), foci (0, 2\sqrt{3}) and (0, -2\sqrt{3}).

Next, plot these key points on the coordinate plane and connect them in a smooth, symmetrical shape, ensuring the curvature fits the verticality or horizontality described by the major axis.
The shape of the ellipse should reflect the relationship between its axes, and in this instance, it's taller than it is wide due to its vertical orientation.
This visualization aids in understanding the fundamental properties of ellipses, while also providing a geometrical context to the algebraic equations involved.