Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertical major axis of length 7 , minor axis of length 6

Step-by-Step Solution

Answer

\( \frac{x^2}{9} + \frac{y^2}{12.25} = 1 \)

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

Given that the major axis length is 7 and it is vertical, this means the semi-major axis is half of this length, so \( a = \frac{7}{2} = 3.5 \). For the minor axis with length 6, the semi-minor axis is \( b = \frac{6}{2} = 3 \).

Step 2: Determine the form of the ellipse equation

Since the ellipse is centered at the origin and has a vertical major axis, the standard form of the ellipse equation is: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]

Step 3: Substitute the values of a and b into the equation

Use the values found in Step 1: \( a = 3.5 \) and \( b = 3 \). Substitute these into the ellipse equation: \[ \frac{x^2}{3^2} + \frac{y^2}{3.5^2} = 1 \]

Step 4: Simplify and finalize the equation

Calculate \( b^2 = 9 \) and \( a^2 = 12.25 \), substitute these into the formula: \[ \frac{x^2}{9} + \frac{y^2}{12.25} = 1 \] This is the final form of the ellipse equation.

Key Concepts

Understanding the Semi-Major Axis

An ellipse is defined by two main axes: the major axis and the minor axis. The **semi-major axis** is half the length of the major axis. This axis is crucial as it determines the longest distance from the center to the ellipse's edge. In our exercise, the major axis has a length of 7 units, meaning the semi-major axis is \( \frac{7}{2} = 3.5 \) units.

• This axis dictates the orientation of the ellipse.
• The longer the semi-major axis compared to the semi-minor axis, the more elongated the ellipse becomes.

In this case, since the major axis is vertical, the semi-major axis also extends vertically from the center of the ellipse.

Exploring the Semi-Minor Axis

The **semi-minor axis** is half the length of the minor axis and it defines the shortest distance from the center to the edge of the ellipse. In our given problem, the minor axis is 6 units in total, so the semi-minor axis is \( \frac{6}{2} = 3 \) units.

• Since the major axis is vertical, the minor axis—and therefore the semi-minor axis—lies horizontally.
• Even though it determines the shorter edge of the ellipse, it is equally important for defining the overall shape.

The semi-minor axis plays a key role in distinguishing the characteristics of the ellipse, alongside the semi-major axis.

Standard Form of the Ellipse Equation

The **standard form of an ellipse equation** helps us represent ellipses algebraically. This form considers whether the major axis is vertical or horizontal. For a vertical major axis, the ellipse equation is written as: \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

• If the major axis were horizontal, the roles of \(a\) and \(b\) would switch in the equation.
• This equation format shows that the squared terms are scaled by their respective semi-axis lengths, ensuring that the ellipse fits neatly within a bounding box of sorts.

Substituting the known values like \(a = 3.5\) and \(b = 3\) helps us derive the specific equation for this ellipse.

Ellipse Center at the Origin

Having the **center at the origin** means the ellipse is centered at the point \((0, 0)\) in the coordinate plane. This simplifies the ellipse equation as it doesn't require additional terms to account for any horizontal or vertical shifts.

• The symmetry of the ellipse about both axes remains intact.
• For an ellipse centered at the origin, the axes of symmetry directly align with the coordinate axes.

Knowing the center at the origin allows us to focus on just the ellipse dimensions— major and minor axes—without additional translation, maintaining a straightforward mathematical representation.