Problem 43 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the ellipse. Identify the foci and vertices. $$ \frac{(x-1)^{2}}{9}+\frac{(y-1)^{2}}{25}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at \((1, 1)\), with vertices \((1, 6)\) and \((1, -4)\), and foci \((1, 5)\) and \((1, -3)\).

Step 1: Identify the General Form of the Ellipse Equation

The given equation \( \frac{(x-1)^2}{9} + \frac{(y-1)^2}{25} = 1 \) is in the form of an ellipse \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Here, \( h = 1 \), \( k = 1 \), \( a^2 = 9 \), and \( b^2 = 25 \).

Step 2: Determine the Center of the Ellipse

The center of the ellipse is given by the point \((h, k)\). Thus, the center of the ellipse is \((1, 1)\).

Step 3: Determine the Semi-Major and Semi-Minor Axes

Since \( b^2 = 25 \) is greater than \( a^2 = 9 \), \( b \) is the semi-major axis. Therefore, \( b = \sqrt{25} = 5 \) and \( a = \sqrt{9} = 3 \). The semi-major axis is vertical, and the semi-minor axis is horizontal.

Step 4: Identify the Vertices of the Ellipse

The vertices along the semi-major axis are at \((h, k \pm b)\). Thus, the vertices are at \((1, 1 \pm 5)\), which are \((1, 6)\) and \((1, -4)\).

Step 5: Identify the Foci of the Ellipse

The foci are given by \((h, k \pm c)\), where \( c = \sqrt{b^2 - a^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \). Therefore, the foci are at \((1, 1 \pm 4)\), which are \((1, 5)\) and \((1, -3)\).

Step 6: Sketch the Ellipse

Based on the determined vertices and foci, sketch the ellipse. Start by plotting the center at \((1, 1)\). Then draw the major and minor axes centered at this point. Place vertices and foci along the vertical direction since it's the major axis. Sketch the shape of the ellipse smoothly around these points.

Key Concepts

Graphing Ellipses

Graphing an ellipse involves a series of methodical steps, allowing us to understand its properties clearly. An ellipse is a conic section and is visually represented as an oval shape on a graph. The equation for an ellipse is usually provided in the form \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]here, - \( (h, k) \) is the center of the ellipse.- \( a \) represents the length of the semi-minor axis, and \( b \) represents the length of the semi-major axis, assuming \( b > a \).
To graph the ellipse, you begin by plotting its center point at \( (h, k) \). From the center, measure \( a \) units horizontally in both directions to points on the ellipse and \( b \) units vertically to locate additional points. These points guide the smooth, oval curve around the center that characterizes the elliptical shape. Once these are plotted, connect the points gently to reflect the continuous, closed nature of the ellipse.

Conic Sections

Conic sections are fundamental curves obtained by intersecting a plane with a double-napped cone. The family of conic sections includes circles, ellipses, parabolas, and hyperbolas. Ellipses, as part of this group, represent configurations where the intersecting plane cuts through, but does not include the apex of the cone. This specific type of cut results in a balanced, oval curve, distinguishing it from other conics.
Conic sections have a symmetric, continuous nature, dictated by various parameters like eccentricity, which describes how much an ellipse deviates from being a circle. For ellipses, the eccentricity is less than 1, unlike circles which have an eccentricity of 0.
Ellipses can appear horizontally oriented or vertically oriented based on the comparison of \( a \) and \( b \). When \( b > a \), the major axis is vertical, giving the ellipse a vertical stretch. If \( a > b \), the ellipse is stretched horizontally.

Vertices and Foci of Ellipses

The vertices and foci are crucial identifiers for any ellipse that provide insight into its shape and structure. These points are aligned along the major axis of the ellipse.
- **Vertices**: The vertices of an ellipse are the points that lie at the extreme ends of its major axis. You calculate them using the center \( (h, k) \)and adding and subtracting \( b \) (if the major axis is vertical) from the \( y \)-coordinate. For a horizontal major axis, \( a \)is added and subtracted from the \( x \)-coordinate.- **Foci**: The foci of an ellipse serve as special points that define its curve. They are calculated similarly but involve another component, \( c \),where \( c = \sqrt{b^2 - a^2} \).The foci are located \( c \)units from the center along the major axis.
These structural markers (vertices and foci) not only guide the drawing of the ellipse but also provide geometric insight into how the ellipse behaves as a conic section. Understanding them allows easier interpretations and calculations related to various geometric applications.