Problem 1 - Thomas Calculus

Question

In Exercises \(1-8,\) find the eccentricity of the ellipse. Then find and graph the ellipse's fociand directrices. $$ 16 x^{2}+25 y^{2}=400 $$

Step-by-Step Solution

Answer

Eccentricity is \(\frac{3}{5}\), foci at \((3, 0), (-3, 0)\), and directrices at \(x = \frac{25}{3}\) and \(x = -\frac{25}{3}\).

Step 1: Convert the Equation to Standard Form

The given equation is \(16x^2 + 25y^2 = 400\). We start by dividing every term by 400 to get the standard form of the ellipse equation. This gives us \(\frac{x^2}{25} + \frac{y^2}{16} = 1\).

Step 2: Identify the Major and Minor Axes

The equation \(\frac{x^2}{25} + \frac{y^2}{16} = 1\) is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 25\) and \(b^2 = 16\). Since \(a^2 > b^2\), the major axis is along the x-axis.

Step 3: Calculate the Eccentricity

The formula for eccentricity \(e\) of an ellipse is \(e = \frac{c}{a}\), where \(c = \sqrt{a^2 - b^2}\). Here \(a = 5\) and \(b = 4\), so \(c = \sqrt{25 - 16} = \sqrt{9} = 3\). Thus, the eccentricity is \(e = \frac{3}{5}\).

Step 4: Determine the Foci

The foci \((\pm c, 0)\) are found using \(c = 3\). Hence, the foci are at \((3, 0)\) and \((-3, 0)\).

Step 5: Find the Directrices

The directrix distance \(d\) from the center for an ellipse is \(d = \frac{a}{e}\). With \(a = 5\) and \(e = \frac{3}{5}\), \(d = \frac{5}{\frac{3}{5}} = \frac{25}{3}\). Thus, the equations of the directrices are \(x = \frac{25}{3}\) and \(x = -\frac{25}{3}\).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse can greatly simplify graphing and analyzing ellipses. We start with a general equation like \[ Ax^2 + By^2 = C, \]where the coefficients A and B might differ depending on the proportions of the ellipse.

• To convert this into a standard form, divide all terms by C.
• This helps us align the equation with the more recognizable structure \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \]where \((h,k)\) is the center of the ellipse.
• In the given problem, transforming \(16x^2 + 25y^2 = 400\) into the standard form involved dividing by 400, resulting in\[ \frac{x^2}{25} + \frac{y^2}{16} = 1. \]
• This indicates that the ellipse is centered at the origin \((0,0)\).

Understanding the standard form allows us to identify important features like axes and orientation quickly.

Major and Minor Axes

The major and minor axes of an ellipse are crucial in understanding its shape and orientation.

• The major axis is the longest diameter, while the minor axis is the shortest diameter.
• In the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the larger denominator represents \(a^2\), and the smaller represents \(b^2\).
• For our ellipse, \(a^2 = 25\) and \(b^2 = 16\), meaning \(a = 5\) and \(b = 4\).
• Since \(a^2 > b^2\), the major axis aligns with the x-axis, spanning from \((-5, 0)\) to \((5, 0)\).
• The minor axis is along the y-axis, ranging from \((0, -4)\) to \((0, 4)\).

These axes define the orientation and extent of the ellipse.

Foci of an Ellipse

The foci are two special points located along the major axis of an ellipse. They are used in defining the ellipse itself.

• The distance of each focus from the center is derived from\[ c = \sqrt{a^2 - b^2}. \]
• For the exercise with \(a = 5\) and \(b = 4\), we compute \(c = \sqrt{25 - 16} = 3\).
• This means the foci are positioned at \((3, 0)\) and \((-3, 0)\) along the x-axis.
• These foci make the ellipse unique: For any point on the ellipse, the sum of the distances to each focus is constant.

Understanding the role of the foci helps in grasping other properties like the ellipse's eccentricity.

Directrices of an Ellipse

The directrices of an ellipse are lines associated with its geometry, often aiding in defining its eccentricity.

• For any ellipse, the directrix is located at a distance \(d = \frac{a}{e}\) from the center, where \(e\) is the eccentricity.
• In our case, \(e = \frac{3}{5}\), so the distance becomes \(\frac{5}{\frac{3}{5}} = \frac{25}{3}\).
• This yields vertical directrices running through \(x = \frac{25}{3}\) and \(x = -\frac{25}{3}\).
• Though not as immediately visible as axes or foci, the directrices influence constructions such as the Dandelin spheres, commonly used in geometry.

The directrices, in concert with the foci, provide a deeper geometric understanding of the ellipse's eccentric features.