Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$ \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 $$

Step-by-Step Solution

Answer

Vertices: (5, 0), (-5, 0); Foci: (3, 0), (-3, 0). The ellipse is horizontal.

Step 1: Identify the Equation Form

The given equation of the ellipse is \( \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 \). This is in the standard form of an ellipse centered at the origin: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). The term with the larger denominator is \( x^{2} \) which indicates a horizontal ellipse.

Step 2: Determine the Values of a and b

From the standard form, we identify \( a^{2} = 25 \) and \( b^{2} = 16 \). Solving for \( a \) and \( b \), we get \( a = \sqrt{25} = 5 \) and \( b = \sqrt{16} = 4 \).

Step 3: Calculate the Foci Distance

For an ellipse, the distance from the center to each focus is \( c \), where \( c^{2} = a^{2} - b^{2} \). Substitute \( a^{2} = 25 \) and \( b^{2} = 16 \) into the equation: \( c^{2} = 25 - 16 = 9 \), thus \( c = \sqrt{9} = 3 \).

Step 4: Find the Vertices

Since the ellipse is horizontal, the vertices are located at \( (\pm a, 0) \). Substitute \( a = 5 \) to find the vertices: \( (5, 0) \) and \( (-5, 0) \).

Step 5: Find the Foci

The foci of the ellipse, located along the x-axis, are at \( (\pm c, 0) \). Using \( c = 3 \), the foci are \( (3, 0) \) and \( (-3, 0) \).

Step 6: Sketch the Graph

To sketch the graph, draw a horizontal ellipse centered at the origin (0, 0) with vertices at (5, 0) and (-5, 0), and co-vertices at (0, 4) and (0, -4). Place the foci at (3, 0) and (-3, 0) within the ellipse to show their locations.

Key Concepts

Vertices

In the context of an ellipse, the vertices are specific, important points located on its longer axis, known as the major axis. For a horizontal ellipse, the vertices are equidistant from the center and lie along the x-axis. This distance from the center to each vertex is represented by the value "a."
In our equation, \( \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 \), we see that \( a^{2} = 25 \). When solved, \( a = \sqrt{25} = 5 \).
Therefore, the vertices are at the points \((5, 0)\) and \((-5, 0)\).
These points provide the maximum horizontal extent from the origin, indicating the overall width of the ellipse along its major axis.

Foci

The foci are special points inside the ellipse that play a crucial role in its geometry. The sum of the distances from any point on the ellipse to each focus is constant, defining the shape uniquely. In a horizontal ellipse, the foci are also positioned along the major axis.
To find the distance \( c \) from the center to each focus, use the equation \( c^{2} = a^{2} - b^{2} \).

• Given \( a^{2} = 25 \) and \( b^{2} = 16 \), solving gives us \( c^{2} = 25 - 16 = 9 \), thus \( c = \sqrt{9} = 3 \).

Therefore, the foci for this horizontal ellipse are located at \((3, 0)\) and \((-3, 0)\).
These points, lying inside the ellipse closer to its center than the vertices, help determine the path of the ellipse's curvature.

Standard form of ellipse

The standard form of an ellipse is critical for identifying and analyzing its properties. It allows us to easily see the orientation and dimensions of the ellipse.
For an equation of the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), the values \( a \) and \( b \) represent the semi-major and semi-minor axes, respectively.
To determine the orientation:

• If \( a^{2} > b^{2} \), the ellipse is horizontal.
• If \( b^{2} > a^{2} \), it is vertical.

This standard form helps identify whether an ellipse is stretched horizontally or vertically, aiding in the rapid identification of its key features, such as vertices and foci.

Horizontal ellipse

A horizontal ellipse is one where the major axis runs along the x-axis. This is indicated when the denominator of the \( x^2 \) term is greater than that of the \( y^2 \) term in the equation of the ellipse.
In our case, the equation is \( \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 \), showing us that \( a^{2} = 25 \) and \( b^{2} = 16 \). With \( a > b \), this confirms it is a horizontal ellipse.
For such ellipses, the vertices lie on the x-axis at \((\pm a, 0)\), and the foci also lie on the x-axis at \((\pm c, 0)\).
This orientation fundamentally affects how the ellipse is graphically represented and determines the axes along which its main features align.