Problem 34 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis vertical with length \(20 ;\) length of minor axis \(=10 ;\) center: \((2,-3)\)

Step-by-Step Solution

Answer

The standard form of the ellipse equation is \(\frac{(x-2)^2}{25} + \frac{(y+3)^2}{100} = 1\)

Step 1: Determining the values of a, b, h, and k

From the exercise, the length of the major axis is 20, thus \(a = \frac{20}{2} = 10\). The length of the minor axis is 10, so \(b = \frac{10}{2} = 5\). The center of the ellipse is given as (2,-3), so \(h = 2\) and \(k = -3\)

Step 2: Substitute the values into the standard form equation

The standard form equation of the ellipse will be \(\frac{(x-2)^2}{5^2} + \frac{(y+3)^2}{10^2} = 1\)

Step 3: Simplify the equation

Simplify the equation to get the standard form of the ellipse equation as \(\frac{(x-2)^2}{25} + \frac{(y+3)^2}{100} = 1\)

Key Concepts

Standard Form Equation

The standard form of an equation of an ellipse is a crucial concept in understanding the geometry and properties of this shape. The equation takes on different forms depending on the orientation of the major axis: horizontal or vertical. For an ellipse centered at

• If the major axis is horizontal, the equation is: \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]
• If the major axis is vertical, the equation changes to: \[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]

In these equations, \(h\) and \(k\) represent the coordinates of the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. In the given problem, we determined that since the major axis is vertical, our equation should match the vertical orientation form. This approach helps us to identify the correct formula swiftly, ensuring that we effectively use the known dimensions of the ellipse - the lengths of the major and minor axes, and the center.

Major Axis

The major axis of an ellipse is the longest diameter that passes through its center, stretching from one side to the other. In simple terms, this is the line that dictates the longest direction of the ellipse.For the given ellipse, we know the major axis is vertical with a total length of 20. To find the semi-major axis, we simply divide this length by two. The result gives us \(a = 10\).

• The major axis running vertically indicates an upward and downward orientation around the center.
• This longest line of the ellipse is crucial for identifying its shape and scale in the vertical direction.

Minor Axis

The minor axis of an ellipse is the shorter diameter running perpendicular to the major axis and passing through the center. It determines how narrow the ellipse is in the opposite direction of the major axis.In our problem, the minor axis length is 10. This length, when halved, gives the semi-minor axis length, so \(b = 5\).

• This axis defines the breadth of the ellipse, making it wider in the direction orthogonal to the major axis orientation.
• The minor axis being shorter is a defining feature that creates the "stretched" or "compressed" appearance of an ellipse compared to a circle.

Center of Ellipse

The center of an ellipse is a pivotal point where both axes intersect. It is defined by coordinates \((h, k)\). The center serves as the origin point for measuring distances along the axes. For this problem, our ellipse is centered at \((2, -3)\), thus \(h = 2\) and \(k = -3\). This means all transformations on the ellipse, such as stretching along the axes, occur around this central point.

• This coordinate position determines how the equation is translated within the coordinate plane.
• Knowing the center allows us to accurately place the ellipse and understand its geometric relationship with other points or lines in the plane.