Problem 18 - Precalculus with Limits

Question

In Exercises 11-18, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertical major axis; passes through the points \((0, 6)\) and \((3, 0)\)

Step-by-Step Solution

Answer

The standard form equation of the ellipse with the given characteristics is: \( \frac{x^2}{9} + \frac{y^2}{36} = 1 \)

Step 1: Identify the semi-major axis length

Since the major axis is vertical, and we know that the ellipse passes through the point (0,6), it means that 6 is the semi-major axis length. Hence, \(b = 6\). We square it to get \(b^2 = 36\).

Step 2: Identify the semi-minor axis length

The ellipse passes through the point (3,0), indicating that 3 is the length of the semi-minor axis. Hence, \(a = 3\). We square it to get \(a^2 = 9\).

Step 3: Write the equation of the ellipse

We substitute \(a^2\) and \(b^2\) into the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) to get \( \frac{x^2}{9} + \frac{y^2}{36} = 1 \). This is the standard form of the equation of the ellipse.

Key Concepts

Standard Form of an Ellipse Equation

The equation of an ellipse in standard form holds the key to unlocking its geometric properties. The standard form for an ellipse centered at the origin (0,0), with a vertical major axis, is given by:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

This equation helps us determine the orientation and dimensions of the ellipse. Here, \( a \) represents the semi-minor axis length, and \( b \) is the semi-major axis length when the major axis is vertical.
Understanding these parameters is crucial because they define the ellipse's span in the x and y directions. By identifying \( a^2 \) and \( b^2 \), you know how far the ellipse stretches horizontally and vertically, respectively.
In the context of this exercise, since we know the ellipse has a vertical major axis, it is essential to assign \( b \) as the longer axis based on the given points.

Semi-Major Axis

The semi-major axis is the longest radius of the ellipse. For ellipses centered at the origin with a vertical major axis, the semi-major axis extends along the y-axis.
The point (0, 6) given in the exercise indicates that the semi-major axis reaches up to 6 units from the center, meaning \( b = 6 \). By squaring this length, we obtain \( b^2 = 36 \).
Understanding the semi-major axis's length plays a significant role in determining the shape and size of the ellipse. It helps you visualize how the ellipse expands more in the vertical direction compared to its horizontal reach.

Semi-Minor Axis

Contrarily, the semi-minor axis is the shorter radius of the ellipse. In cases where the ellipse has a vertical major axis, the semi-minor axis will spread out along the x-axis.
From the exercise, the point (3, 0) helps us derive that the semi-minor axis is 3 units long, leading to \( a = 3 \) and thus \( a^2 = 9 \).
The semi-minor axis effectively dictates the breadth of the ellipse in the horizontal direction. While it is shorter than the semi-major axis, its measurement is crucial to form the standard ellipse equation correctly.