Problem 8 - 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. $$ 4 x^{2}+25 y^{2}=100 $$

Step-by-Step Solution

Answer

The vertices are (5,0) and (-5,0); foci are (±√21,0); eccentricity is approximately 0.92; major/minor axes lengths are 10/4.

Step 1: Write the equation in standard form

To convert the given ellipse equation \(4x^2 + 25y^2 = 100\) into its standard form, divide every term by 100. This gives us \(\frac{x^2}{25} + \frac{y^2}{4} = 1\).

Step 2: Identify the major and minor axes

The standard form \(\frac{x^2}{25} + \frac{y^2}{4} = 1\) shows that \(a^2 = 25\) and \(b^2 = 4\). Since \(a^2 > b^2\), this ellipse is horizontal, meaning the major axis is along the x-axis with \(a = 5\) and the minor axis is along the y-axis with \(b = 2\).

Step 3: Find the vertices

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

Step 4: Calculate the foci

For ellipses, the relationship is \(c^2 = a^2 - b^2\). Substituting the values gives \(c^2 = 25 - 4 = 21\), so \(c = \sqrt{21}\). The foci are at \((\pm \sqrt{21}, 0)\), which are approximately \((±4.58, 0)\).

Step 5: Compute the eccentricity

The eccentricity \(e\) of an ellipse is given by \(e = \frac{c}{a}\). So, \(e = \frac{\sqrt{21}}{5} \approx 0.92\).

Step 6: Determine the lengths of the major and minor axes

The length of the major axis is \(2a = 10\) and the length of the minor axis is \(2b = 4\).

Step 7: Sketch the graph

Since it's a horizontal ellipse centered at the origin with semi-major axis \(a = 5\) and semi-minor axis \(b = 2\), plot the ellipse crossing x-axis at ±5 and y-axis at ±2. Add the foci at ±\(\sqrt{21}\) along the x-axis.

Key Concepts

Vertices of an Ellipse

In an ellipse, the vertices are the points where the ellipse intersects its major axis. For the equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), we identify that the ellipse is horizontal because \(a^2 = 25\) is greater than \(b^2 = 4\). This means the major axis lies along the x-axis. The standard form tells us the semi-major axis length \(a\) is 5. Consequently, the vertices, which are located at the endpoints of the major axis, are at \((\pm a, 0) = (5, 0)\) and \((-5, 0)\).

• Vertices give insight into the orientation and size of the ellipse.
• They are important in sketching the ellipse accurately on a graph.

Foci of an Ellipse

The foci of an ellipse are two unique points inside the ellipse. The total distance from these points to any point on the ellipse remains constant. The relationship \(c^2 = a^2 - b^2\) helps us find these points in the standard form equation \(\frac{x^2}{25} + \frac{y^2}{4} = 1\). Here, \(a = 5\) and \(b = 2\), thus \(c^2 = 25 - 4 = 21\) and \(c = \sqrt{21}\).The foci are along the major axis (since the ellipse is horizontal) at \((\pm \sqrt{21}, 0)\), approximately \((\pm 4.58, 0)\).

• Foci play a crucial role in defining the ellipse's shape.
• Located on the major axis, they are always inside the ellipse.

Eccentricity of an Ellipse

Eccentricity, represented by \(e\), measures how "stretched" an ellipse is. It is the ratio of the distance between a focus and the center to the semi-major axis. For our ellipse, calculate the eccentricity using the formula \(e = \frac{c}{a}\). For values we already derived: \(c = \sqrt{21}\) and \(a = 5\). This results in \(e = \frac{\sqrt{21}}{5} \approx 0.92\). The closer the \(e\) value is to 1, the more elongated the ellipse.

• Eccentricity differentiates an ellipse from a circle (where \(e = 0\)).
• Helps gauge how close the ellipse is to being a perfect circle.

Major and Minor Axes

The major and minor axes of an ellipse are the longest and shortest diameters, respectively. For the ellipse \(\frac{x^2}{25} + \frac{y^2}{4} = 1\), the major axis is along the x-axis.

• The length of the major axis is \(2a\), calculated as \(2 \times 5 = 10\).
• The minor axis runs along the y-axis, with a length of \(2b\), giving \(2 \times 2 = 4\).

The axes are crucial in describing the complete dimensions of the ellipse, ensuring accuracy in graph representation.

Standard Form of an Ellipse

The standard form of an ellipse equation is a way to represent its algebraic equation. It emphasizes the orientation and dimensions of the ellipse. To rewrite the given equation \(4x^2 + 25y^2 = 100\) into standard form, divide every term by 100 to get \(\frac{x^2}{25} + \frac{y^2}{4} = 1\).

• This form highlights whether the ellipse is horizontal or vertical based on whether \(a^2\) or \(b^2\) is larger.
• Helps easily identify key properties such as the axes' lengths and direction.

It acts as a foundation to calculate and visualize other aspects like vertices and eccentricity.