Problem 46 - Algebra 2

Question

Write an equation for each conic section. Then sketch the graph. $$ (5,-3), \text { one focus at }(5,0), \text { and one vertex at }(5,-1) $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{(x-5)^2}{9} + \frac{(y+3)^2}{4} = 1\). The graph is a vertical ellipse centered at (5,-3), with foci at (5,0) and (5,-6), and vertices at (5,-1) and (5,-5)

Step 1: Find the semi-major and semi-minor axes

The semi-major axis, \(a\), can be found by measuring the distance from the center of the ellipse to the vertex. The coordinates of the center are (5,-3) and of a vertex are (5,-1). So \(a = |-3 - (-1)| = 2\). The semi-minor axis, \(b\), can be found by measuring the distance from the center to a focus. The coordinates of the center are (5,-3) and the coordinates of a focus are (5,0). So \(b = |-3 - 0| = 3\). Thus, we can conclude that \(a=2\) and \(b=3\).

Step 2: Form the equation of the ellipse

The standard form for the equation of a vertical ellipse with center at \((h, k)\) is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\). Plugging the known values: \(h=5\), \(k=-3\), \(a=2\), \(b=3\), the equation is: \(\frac{(x-5)^2}{3^2} + \frac{(y+3)^2}{2^2} = 1\) or simply \(\frac{(x-5)^2}{9} + \frac{(y+3)^2}{4} = 1\).

Step 3: Sketch the graph

The graph will be an ellipse at the coordinate point (5, -3) (the center), with a length along the major axis (vertical) of 2a=4 and a length along the minor axis (horizontal) of 2b=6. There will be foci at (5, 0) and (5, -6) and vertices at (5, -1) and (5, -5).

Key Concepts

Ellipse

An ellipse is a geometric shape resembling an elongated circle. It has two main axes: the major axis, which is the longest diameter, and the minor axis, the shortest. The defining characteristic of an ellipse is the sum of the distances from any point on its surface to two fixed points, called foci, which are constant. This distinct feature differentiates an ellipse from other conic sections such as hyperbolas and parabolas.
Ellipses are found in everyday life. For example:

• Planets orbit the sun in an elliptical path.
• Elliptical shapes are used in architecture and design.

Understanding its properties, like its symmetry and central location, is crucial to solving conic section problems.

Focus

The focus of an ellipse is one of the two special points used to define and construct it. In an ellipse, these points are located inside the shape along the major axis. The distance from these foci to any point on the ellipse's perimeter remains consistent when summed. This constant distance is vital for understanding and writing the equation of an ellipse.
To comprehend this concept better, consider this:

• In the exercise, one focus is at the point (5,0).
• Foci are symmetrical, so there is another focus at (5,-6).

Understanding the location and role of the foci will aid in graphing and analyzing the properties of ellipses in various problems.

Vertex

Vertices are crucial points in the geometry of an ellipse as they lie at the ends of the major axis. In simpler terms, the vertices are the 'tips' of the ellipse, showcasing its maximum width or height depending on its orientation. Identifying the vertices is essential for determining the semi-major axis length and hence understanding the scale and orientation of the ellipse.
In our specific exercise:

• One vertex is located at (5,-1).
• The other symmetrical vertex is at (5,-5).

Knowing where these points are helps accurately sketch the ellipse and understand its proportions and orientation in the coordinate plane.

Equation of Ellipse

The equation of an ellipse is derived from its geometric properties, mainly its center, axes, and foci. The standard form for vertical ellipses, where the major axis is vertical, is:\[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]Here,

• \( (h, k) \) represents the center of the ellipse.
• \( a \) is the semi-major axis length, the vertical distance from center to vertex.
• \( b \) is the semi-minor axis length, the horizontal distance from center to focus.

In our exercise, with center \((5,-3)\), semi-major axis \(a=2\), and semi-minor axis \(b=3\), the equation simplifies to:\[ \frac{(x-5)^2}{9} + \frac{(y+3)^2}{4} = 1 \]This equation provides a mathematical description of our specific ellipse.

Semi-major Axis

The semi-major axis of an ellipse is half of its longest diameter. It runs through the center and both vertices. The length of this axis is denoted by \( a \), and it is the key measure in determining the scale of the ellipse.In the ellipse with vertical orientation:

• The semi-major axis is the distance from the center to the vertex along the longer, vertical axis.
• In our example, the center is \((5,-3)\) and a vertex is \((5,-1)\), making \(a=2\).

Understanding the semi-major axis is vital for sketching the ellipse accurately and applies to both determining its size and writing its equation.