Problem 36 - Algebra and Trigonometry

Question

In Exercises \(25-36,\) find the standard form of the equation of each ellipse satisfying the given conditions. Endpoints of major axis: \((2,2)\) and \((8,2)\) Endpoints of minor axis: \((5,3)\) and \((5,1)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \(\frac{(x-5)^2}{9} + (y-2)^2 = 1\).

Step 1: Finding the Center

To find the center of the ellipse, calculate the midpoints of either the major or minor axis. Given the endpoints of the major axis as \( (2,2) \) and \( (8,2) \), in this case, the center would be ((2+8)/2 , (2+2)/2) = (5,2).

Step 2: Finding the Semi-major and Semi-minor Axises'

The length of the semi-major axis (a) and semi-minor axis (b) are half of the length of the major and minor axis respectively. Given the endpoints of the major axis \((2,2)\) and \((8,2)\), the length of the major axis is |8-2| = 6. So, the semi-major axis is 6/2 = 3. Similarly, for the minor axis, given the endpoints as \((5,3)\) and \((5,1)\), the length of the minor axis is |3 - 1| = 2. So, the semi-minor axis (b) is 2/2 = 1.

Step 3: Writing the Standard Form of the Equation

The standard form of the equation of an ellipse centered at \((h,k)\) with a horizontal major axis is given by \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). Replacing \(h=5, k=2, a=3\), and \(b=1\), we obtain \(\frac{(x-5)^2}{3^2} + \frac{(y-2)^2}{1^2} = 1\).

Key Concepts

Standard Form of Ellipse Equation

Understanding the equation of an ellipse in its standard form is fundamental to solving problems related to this shape. Ellipses, like circles, have a standard equation that represents all points on the curve. The standard form depends on whether the ellipse is horizontal (major axis along the x-axis) or vertical (major axis along the y-axis).

For a horizontal ellipse, the general standard form is:
\[\begin{equation} \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\end{equation}\]
where \( h \) and \( k \) are the coordinates of the center of the ellipse, \( a \) is the length of the semi-major axis, and \( b \) is the length of the semi-minor axis. The variable \( x \) represents any x-coordinate on the ellipse, and \( y \) represents any y-coordinate.

Conversely, for a vertical ellipse, the form is slightly different:
\[\begin{equation} \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\end{equation}\]
Here, the roles of \( a \) and \( b \) are switched due to the orientation of the axes.

In the given example, the semi-major axis is horizontal. Thus, based on the end points given and the midpoint calculated, the standard form equation for our ellipse becomes: \[\begin{equation} \frac{(x-5)^2}{3^2} + \frac{(y-2)^2}{1^2} = 1\end{equation}\]
This equation is derived using the distances between the center and the vertices along both axes, providing the respective squares of \( a \) and \( b \) under their corresponding variable's squared difference in the equation.

Semi-Major and Semi-Minor Axes of an Ellipse

In every ellipse, the semi-major and semi-minor axes are key to understanding its shape and dimensions. The semi-major axis, \( a \), is half the distance across the longest part of the ellipse, while the semi-minor axis, \( b \), is half the distance across the shortest part.

It's easy to remember: 'major' means larger, so the semi-major axis will always be equal to or longer than the semi-minor axis. These axes provide critical information when determining the standard form of an ellipse's equation, as seen in the exercise.

By finding the length between the given endpoints of the major and minor axes and halving them, we determined the lengths of the semi-major axis, \( a = 3 \), and the semi-minor axis, \( b = 1 \), for the described ellipse. This measurement is essential to plug into the standard form equation to describe our ellipse accurately.

Finding the Center of an Ellipse

To find the center of an ellipse, you simply need to locate the midpoint between the endpoints of either the major or the minor axis. The center of an ellipse is the point from which the distances to the curve remain constant along the axes, and it serves as \( (h,k) \) in the standard equation of an ellipse.

In our exercise, the given endpoints of the major axis were \( (2,2) \) and \( (8,2) \). To find the midpoint, we averaged the x-coordinates and the y-coordinates separately: \[\begin{equation} h = \frac{2 + 8}{2}, k = \frac{2 + 2}{2}\end{equation}\]
The midpoint, or the center of our ellipse, is \( (5,2) \). Knowing the center is the beginning step to formulating the rest of the standard equation of an ellipse, anchoring it in the coordinate plane and setting us up to explore its dimensions thoroughly.