Problem 10 - Algebra and Trigonometry

Question

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ \frac{x^{2}}{16}+\frac{y^{2}}{25}=1 $$

Step-by-Step Solution

Answer

Vertices at (0, ±5), foci at (0, ±3), eccentricity 0.6, major axis 10, minor axis 8.

Step 1: Identify the Standard Form of the Ellipse

The equation given is \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \). This is in the standard form of an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( b^2 > a^2 \).

Step 2: Determine the Orientation

Since \( b^2 = 25 \) is greater than \( a^2 = 16 \), the ellipse is vertical. So, \( b = 5 \) and \( a = 4 \).

Step 3: Find the Vertices

For a vertical ellipse, the vertices are at \((0, \pm b)\). Thus, the vertices are at \((0, 5)\) and \((0, -5)\).

Step 4: Calculate the Foci

The distance between the center and each focus is \( c = \sqrt{b^2 - a^2} = \sqrt{25 - 16} = \sqrt{9} = 3 \). The foci are at \((0, \pm c)\), or \((0, 3)\) and \((0, -3)\).

Step 5: Find the Eccentricity

The eccentricity \( e \) of the ellipse is given by \( e = \frac{c}{b} = \frac{3}{5} = 0.6 \).

Step 6: Determine the Lengths of Major and Minor Axes

The length of the major axis is \(2b = 10\), and the length of the minor axis is \(2a = 8\).

Step 7: Sketch the Graph

Since it is a vertical ellipse centered at the origin with vertices at (0, 5) and (0, -5), and minor axis endpoints at (4, 0) and (-4, 0), sketch the ellipse on a coordinate plane with these dimensions.

Key Concepts

Vertices of Ellipse

The vertices of an ellipse are special points where the ellipse stretches the furthest in its major direction. For the ellipse described by the equation \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), the vertices can be found directly from the values of \(a\) and \(b\). Since we have a vertical ellipse (because \(b^2 > a^2\)), the vertices are located on the y-axis.
Their coordinates are determined by \((0, \pm b)\). Therefore, you find the vertices of the ellipse at \((0, 5)\) and \((0, -5)\). These points show how, in the y-direction, the ellipse reaches its maximum positive and negative extents.
To remember:

• In a horizontal ellipse: vertices are \((\pm a, 0)\).
• In a vertical ellipse: vertices are \((0, \pm b)\).

Foci of Ellipse

The foci of an ellipse are two fixed interior points, which play a crucial role in its geometric definition. The sum of the distances from any point on the ellipse to the two foci is constant.
For the ellipse \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), you calculate the distance from the center of the ellipse to each focus using the formula \(c = \sqrt{b^2 - a^2}\). Here, it calculates to \(c = 3\).

• For a vertical ellipse (\(b^2 > a^2\)), the foci are at \((0, \pm c)\).
• For a horizontal ellipse, the foci lie on the x-axis.

You find the foci at the points \((0, 3)\) and \((0, -3)\). These points help define the shape of the ellipse.

Eccentricity of Ellipse

Eccentricity is a number between 0 and 1 that measures how "stretched" an ellipse is compared to a circle, where 0 indicates a perfect circle.
For this ellipse, the eccentricity \(e\) is calculated as \(e = \frac{c}{b}\). So, with \(c = 3\) and \(b = 5\), the eccentricity is \(0.6\). This tells us that the ellipse is moderately elongated, neither too stretched nor too close to a circle.

• If \(e \to 0\), the ellipse is more circular.
• If \(e \to 1\), the ellipse becomes more elongated.

Eccentricity is a key identifier in distinguishing ellipses from circles.

Major and Minor Axes of Ellipse

The major and minor axes of an ellipse are its longest and shortest diameters, respectively. They intersect at the center of the ellipse, forming a right angle.
In the equation \(\frac{x^2}{16} + \frac{y^2}{25} = 1\), we identify that because \(b^2 > a^2\), the major axis is vertical and described by its full length \(2b = 10\). Therefore, the vertical span of the ellipse through its center is 10 units.
The minor axis, being horizontal, measures \(2a = 8\). This means from end to end across its narrowest width, the ellipse covers 8 units in the horizontal direction.
Understanding these axis lengths helps visualize and draw the basic dimensions of the ellipse.

Sketching Ellipses

Sketching an ellipse can initially seem challenging, but breaking it down into steps makes it manageable. Follow this guideline for the ellipse equation \(\frac{x^2}{16} + \frac{y^2}{25} = 1\):

• Plot the center at the origin: \((0,0)\).
• Mark the vertices at \((0,5)\) and \((0,-5)\).
• Identify the endpoints of the minor axis at \((4,0)\) and \((-4,0)\).
• Draw smooth curves to connect these points in an oval shape.

The ellipse is vertical due to its major axis being aligned with the y-axis. Remember to use a light hand to ensure smooth and proportionate quadrants. Once the basic shape is sketched, you can darken your lines for emphasis.
Sketching is a crucial skill for visualizing complex mathematical relations and helps consolidate your understanding of an ellipse's properties.