Problem 33 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length \(10 ;\) length of minor axis \(=4\) center: \((-2,3)\)

Step-by-Step Solution

Answer

The standard equation of the ellipse is \((x+2)^2/4 +(y-3)^2/25 =1 \).

Step 1: Identify Values

Firstly, calculate 'a' and 'b'. In this case, \(a\) is half the length of the major axis and \(b\) is half the length of the minor axis. So, \(a=10/2=5\) and \(b=4/2=2\). The center \((h, k)\) is given as \((-2,3)\).

Step 2: Substitute the Values into the Equation

Replace \(a\), \(b\), \(h\), and \(k\) in the equation of the ellipse. Therefore, \((x-(-2))^2/2^2 +(y-3)^2/5^2=1 \) which simplifies to \((x+2)^2/4 +(y-3)^2/25 =1 \).

Key Concepts

Standard Form of an Ellipse

Understanding the standard form of an ellipse is crucial for solving related problems in geometry and algebra. Essentially, the standard form of an ellipse equation is written as \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) for horizontal ellipses, and \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\) for vertical ellipses. Here, \(a\) and \(b\) denote the lengths of the semi-major and semi-minor axes, respectively. The center of the ellipse is denoted by the coordinates \((h, k)\).

To solve the equation for the standard form, first calculate the values of \(a\) and \(b\). Remember that \(a\) is the radius along the major axis, and \(b\) is the radius along the minor axis. Next, identify the center of the ellipse \((h, k)\) to replace the variables in the equation with the corresponding values, thereby obtaining the standard form of the ellipse.

Major and Minor Axes of an Ellipse

The major and minor axes are key features of an ellipse. The major axis is the longest diameter that passes through the center of the ellipse, and the minor axis is the shortest diameter, also passing through the center. The lengths of these segments are pivotal to the shape and size of the ellipse.

In the given problem, we have the lengths of the major and minor axes provided. To find the semi-axes lengths, we halve these values. Therefore, for this ellipse, the semi-major axis \(a\) is 5 units long (half of 10), and the semi-minor axis \(b\) is 2 units long (half of 4).

Importance of Semi-Axes

The semi-major and semi-minor axes not only determine the shape of the ellipse but are also used in calculating the foci and the eccentricity of the ellipse, which are important properties in understanding its geometry.

Coordinates of Ellipse Center

The ellipse's center is a fixed point from which the distances to any point on the ellipse are governed by a constant sum. The coordinates \((h, k)\) define the location of the center on the Cartesian plane. It is important to accurately identify this point, as it will be used in plotting the ellipse and in various calculations involving the ellipse.

For the exercise provided, the center of the ellipse is given as coordinates \((-2,3)\). This means that \(h=-2\) and \(k=3\). These coordinates indicate that the center is located two units to the left and three units up from the origin of the coordinate system. Always remember that the center is pivotal in graphing the ellipse, as all points on the ellipse are equidistant from the center based on the lengths of their respective axes.