Problem 63 - Algebra 2

Question

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. \(a=2 \sqrt{5}, b=3 \sqrt{2},\) width 6\(\sqrt{2}\)

Step-by-Step Solution

Answer

The equation of the ellipse in standard form with the given characteristics is \(x^2/20 + y^2/18 = 1\).

Step 1: Identify the values of \(a\) and \(b\)

From the problem, the semi-major axis \(a = 2\sqrt{5}\) and the semi-minor axis \(b = 3\sqrt{2}\).

Step 2: Substitute in the Ellipse Formula

The formula for an ellipse with center at the origin is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substituting values of \(a\) and \(b\), the equation of the ellipse becomes \(\frac{x^2}{(2\sqrt{5})^2} + \frac{y^2}{(3\sqrt{2})^2} = 1\)

Step 3: Simplification

Simplify the equation to its simplest form by performing the operations in the denominator will lead to \(x^2/20 + y^2/18 = 1\)

Key Concepts

Semi-Major Axis

The semi-major axis of an ellipse is the longest radius of the ellipse, stretching from the center to the farthest point on the edge. In our exercise, this length is denoted by \(a\), and is given as \(2\sqrt{5}\).

• The semi-major axis is crucial in determining the elongated shape of the ellipse.
• It always lies along the plane of symmetry and represents the longest distance from the center.
• In many problems, it is also used to measure how stretched the ellipse is compared to a circle.

To find the squared length \(a^2\), simply square \(2\sqrt{5}\), resulting in \(20\). This value plays an important part in the denominator of the ellipse's equation in standard form.

Semi-Minor Axis

The semi-minor axis is the shortest radius of the ellipse, extending from the center to the closest point on the edge. In our example, the semi-minor axis \(b\) has the value of \(3\sqrt{2}\).

• This axis is always perpendicular to the semi-major axis.
• Its length defines the narrowest width of the ellipse, indicating how "compressed" the ellipse looks.

To compute \(b^2\), which is used in the ellipse formula, square \(3\sqrt{2}\) to get \(18\). The smaller the semi-minor axis, the thinner or more elongated the ellipse appears.

Ellipse Formula

An ellipse in its most basic form resembles a stretched circle. The general ellipse formula for an ellipse centered at the origin is given by: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

• This formula holds that the sum of the squares of the distances from any point on the ellipse to the two focal points is constant.
• It organizes the coordinates \(x\) and \(y\) based on the semi-major and semi-minor axes, \(a\) and \(b\).

In our exercise, substituting the given \(a\) and \(b\) values transforms the formula into \(\frac{x^2}{20} + \frac{y^2}{18} = 1\). This formula effectively describes the unique size and orientation of the ellipse based on the given dimensions.

Standard Form

The standard form of an ellipse equation is designed to highlight its geometric properties. An ellipse centered at the origin has its standard form expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

• This form is especially useful in algebra to clearly outline the relationship between the ellipse's dimensions and its positioning relative to the axes.
• Knowing \(a\) and \(b\) enables easy computation and graphing of the ellipse.
• It acts as a fundamental model in conic sections, providing a base for modifying and comparing ellipses.

For our specific problem with values \(a = 2\sqrt{5}\) and \(b = 3\sqrt{2}\), the standard form simplifies to \(\frac{x^2}{20} + \frac{y^2}{18} = 1\). This final equation effectively encapsulates all relevant details of the ellipse within a single, simplified expression.