Problem 36 - Algebra 2

Question

Find the foci for each equation of an ellipse. $$ 25 x^{2}+4 y^{2}=100 $$

Step-by-Step Solution

Answer

The foci of the ellipse are located at points \((0,\sqrt{21})\) and \((0,-\sqrt{21})\).

Step 1: Convert Equation into Standard Form

Begin by dividing each term in the given equation by 100 to convert it into the standard form of an ellipse's equation. This yields \[ \frac{x^{2}}{4} +\frac{y^{2}}{25} = 1 \]

Step 2: Identify the Semi-Major and Semi-Minor Axes

From the standard form of the ellipse equation, we see that the semi-major axis, \(a\), is \(\sqrt{25} = 5\), and the semi-minor axis, \(b\), is \(\sqrt{4} = 2\).

Step 3: Calculate the Distance to the Foci

To find the foci of the ellipse, we calculate the distance of each focus from the center using the equation \(c = \sqrt{a^{2} - b^{2}} = \sqrt{25 - 4} = \sqrt{21}\). The foci are located at the points \((0,\sqrt{21})\) and \((0,-\sqrt{21})\)

Key Concepts

Foci of Ellipse

In an ellipse, the foci are two special points located along the major axis. The property that makes these points unique is that the sum of the distances from any point on the ellipse to the two foci is constant. This characteristic is what defines the shape of the ellipse—it is a closed curve, wider than a circle, and typically resembles an elongated circle or an oval.
To find the foci of an ellipse, we use the formula \( c = \sqrt{a^2 - b^2} \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. The distance \( c \) is the distance from the center of the ellipse to each of the foci. Depending on how the ellipse is oriented, the full coordinates of the foci can be calculated. Generally, they will be \((c, 0)\) and \((-c, 0)\) if stretched along the x-axis, or \((0, c)\) and \((0, -c)\) if along the y-axis.

Semi-Major Axis

The semi-major axis of an ellipse is one of its principal defining characteristics. It represents half the length of the longest diameter of the ellipse.
In an ellipse formatted horizontally, the semi-major axis is the horizontal segment, while for a vertical ellipse, it is the vertical one.

• The value of the semi-major axis, \( a \), comes from the equation of the ellipse in standard form, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
• The semi-major axis determines how "stretched" the ellipse is.

For our ellipse with equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \), the semi-major axis is 5. This corresponds to the \( \sqrt{25} \) derived from the denominator of the ellipse equation.

Semi-Minor Axis

The semi-minor axis is the shortest radius of the ellipse. It measures half the shortest diameter.

• The semi-minor axis is perpendicular to the semi-major axis and helps define the boundary of the ellipse.
• Accessing the value of the semi-minor axis in the standard form of the ellipse is done by examining the smaller denominator \( b^2 \) in \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

For the equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \), the semi-minor axis is 2, corresponding to \( \sqrt{4} \). This signifies a vertical radius shorter than the horizontal radius (semi-major axis).
The relationship between the semi-major and semi-minor axes explains the elongation of the ellipse.

Standard Form of Ellipse

The standard form of an ellipse is a way to represent its equation such that the main features (axes, center, and orientation) can be easily recognized.

• This form is expressed as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) for an ellipse centered at the origin (0, 0).
• The denominators \( a^2 \) and \( b^2 \) are key. They relate to the lengths of the semi-major and semi-minor axes respectively.

By converting any arbitrary equation into its standard form, mathematical analysis and graph representations become significantly more straightforward. For example, for the equation \( 25x^2 + 4y^2 = 100 \), dividing through by 100 yields \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \), establishing the ellipse in a standard form for easy further calculations.

Distance to Foci

The distance to the foci from the center of the ellipse determines positioning and is crucial for understanding the particular layout of the ellipse's geometry.

• The distance, denoted as \( c \), is evaluated using the relation \( c = \sqrt{a^2 - b^2} \) derived from the semi-major \( a \) and semi-minor \( b \) axes lengths.
• This distance helps confirm the placement of foci which delineates how the curve will accommodate or adjust around these central points.

For our specific example, with axes \( a = 5 \) and \( b = 2 \), the foci are \( \pm \sqrt{21} \) units from the origin. These calculations make determining the shape and eccentricity of the ellipse tangible and intuitive.