Problem 25 - Precalculus

Question

Find the standard form of the equation of the ellipse with the given characteristics. Center: (1,3)\(;\) vertex: (-2,3)\(;\) minor axis of length 4

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \((x-1)^2/9 + (y-3)^2/4 = 1\).

Step 1: Identify Center, Vertex and Minor Axis of the Ellipse

The center of the ellipse is given as (1,3), one of the vertices as (-2,3), and the length of the minor axis is 4. We recall that the distance from the center to a vertex of the ellipse is the semi-major axis (a). Thus, we first calculate a by determining the distance between the center and the vertex. Additionally, as the minor axis is given as the full length, the semi-minor axis (b) is half the length of minor axis.

Step 2: Calculate Semi-Major and Semi-Minor Axes

Calculate the semi-major axis 'a' using the formula for distance between two points in a plane: \(a= \sqrt{(x2-x1)^2 + (y2-y1)^2}\) where (x1, y1) is the center and (x2, y2) is the vertex, so \(a= \sqrt{(-2-1)^2 + (3-3)^2} = 3\). The semi-minor axis 'b' is half the length of the minor axis, so \(b= 4/2 = 2\).

Step 3: Write the Standard Form of the Ellipse Equation

Substitute the values of the center coordinates, semi-major axis 'a', and the semi-minor axis 'b' into the standard form of the ellipse equation. Since the center-to-vertex line is horizontal (both have the same y-coordinate), the formula to be used is \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\), where \((h,k)\) are the coordinates of the center. Substituting the values, we get \((x-1)^2/3^2 + (y-3)^2/2^2 = 1\).

Step 4: Simplify the equation

Simplify the equation to get the final answer. \((x-1)^2/9 + (y-3)^2/4 = 1\).

Key Concepts

Semi-Major Axis

The semi-major axis is a key element of an ellipse, resembling the 'longest radius' of the oval shape. It extends from the center to the furthest point on the curve of the ellipse. In relation to the given problem, the semi-major axis is calculated using the distance formula between two points, which in this case is between the center (1,3) and a vertex (-2,3). The coordinates provide a horizontal line, meaning that there is no change in the y-coordinate.

The calculation of the semi-major axis 'a' would be as follows:
\[a = \sqrt{(-2 - 1)^2 + (3 - 3)^2} = \sqrt{9 + 0} = 3.\]
Understanding that the semi-major axis represents half the length of the longest diameter of the ellipse, you can visualize this as the 'width' of the ellipse when the long side is horizontal.

Semi-Minor Axis

In contrast to the semi-major axis, the semi-minor axis of an ellipse is analogous to the 'shortest radius'. It spans from the center to the closest edge of the ellipse. The exercise provides the total length of the minor axis as 4, which represents the diameter. Therefore, to arrive at the semi-minor axis 'b', we simply divide this value by 2, resulting in a semi-minor axis of 2.

\[ b = \frac{4}{2} = 2. \]
This semi-minor axis denotes half the length of the shortest diameter of the ellipse, which can be seen as the 'height' of the ellipse when it is situated with the major axis horizontally.

Ellipse Geometry

The geometry of an ellipse can be thought of as a stretched circle with two distinct radii, the semi-major and semi-minor axes, which determine its shape and size. The standard form of an ellipse's equation reflects this geometry and is expressed as
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \]
where 'a' is the length of the semi-major axis, 'b' is the length of the semi-minor axis, and \((h,k)\) are the coordinates of the ellipse's center. In this form, the ellipse's shape is determined by the ratio of \(a^2\) to \(b^2\), while its location in the coordinate plane is set by the center \((h,k)\).

For the ellipse described in the exercise, the horizontal orientation (as 'a' is greater than 'b') dictates that the semi-major axis is along the x-axis, while the semi-minor axis is along the y-axis. This orientation is crucial when inputting the values into the equation, as it dictates the placement of 'a' and 'b' with respect to the x and y coordinates.

Coordinates of the Ellipse Center

The center of an ellipse plays a pivotal role, serving as the midpoint from which the semi-major and semi-minor axes radiate. For the solved problem, the coordinates of the ellipse center are given as (1,3). These coordinates are vital components in constructing the ellipse's standard equation. They are denoted by 'h' and 'k' in the equation and represent a shift from the origin on the x and y axes respectively.

In the context of our specific equation, \((h, k) = (1, 3)\). Thus, every point on the ellipse maintains a fixed relationship to this center. Knowing the coordinates of the center aids in graphing the ellipse and further understanding its position in the coordinate plane.