Problem 29 - Precalculus

Question

Find the standard form of the equation of the ellipse with the given characteristics. $$\text { Center: }(3,2) ; a=3 c ; \text { foci: }(1,2),(5,2)$$

Step-by-Step Solution

Answer

The equation of the ellipse in standard form is \((x-3)^2/36 + (y-2)^2/8 = 1\)

Step 1: Understand the given information

We know that the formula for an ellipse in standard form is \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\). Here (h, k) is the center. Also, c is the distance from the center to each focus, and a and b are the distances from the center to the points on the ellipse along the major and minor axes respectively.

Step 2: Determine the values of c, a, b

We are given that the center is (3,2) and foci are (1,2) and (5,2). We can determine c by calculating the absolute difference between x values of foci and center, c = |5-3| = 2. Also we are given that a = 3c, so a = 3*2 = 6. The relationship between a, b, c in an ellipse can be defined by \(a^2 = c^2 + b^2\), so \(b = \sqrt{a^2 - c^2} = \sqrt{36-4} = \sqrt{32}\)

Step 3: Substitute values into the standard form

Substitute the above found values into the standard equation of the ellipse, we get \((x-3)^2/6^2 + (y-2)^2/\sqrt{32}^2 = 1\) or simplifying we get \((x-3)^2/36 + (y-2)^2/8 = 1\)

Key Concepts

Ellipse Characteristics

The ellipse is a closed curve on the plane that results from slicing a cone at an angle that is parallel to its base. It can be thought of as a stretched circle with two symmetric axes. These axes are known as the major and minor axes. The points on the ellipse are at a constant sum of distances from two fixed points known as the foci. The general equation for an ellipse centered at \((h, k)\) is \((x-h)^2/a^2 + (y-k)^2/b^2 = 1\), where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

One key characteristic of an ellipse is its eccentricity, which measures the degree of deviation from being circular. The closer the eccentricity is to zero, the more circular the ellipse. Conversely, as it approaches one, the ellipse becomes more elongated. Understanding these basic characteristics of an ellipse is essential for graphing and analyzing its geometrical properties, such as area, focal length, and the perimeter.

Distance from Center to Focus

An important concept when dealing with ellipses is the distance from the center to a focus, denoted as \(c\). Imagine an ellipse as being 'pulled' from a circle from two opposite sides. The points where the 'force' is applied are the foci of the ellipse. The line segment that connects the two foci goes through the center of the ellipse and is of length \(2c\).

The value of \(c\) is related to the lengths of the semi-major and semi-minor axes of the ellipse through the equation \(a^2 = b^2 + c^2\). It's essential to grasp how the semi-major axis \(a\) is always the longest between the two axes, and hence \(c\) will always be less than \(a\) for an ellipse. The distance \(c\) helps determine the shape of the ellipse: the larger \(c\) is in relation to \(a\), the more elongated the ellipse. When \(c\) is zero, the figure is not an ellipse but a circle.

Major and Minor Axes of an Ellipse

The axes of an ellipse are its longest and shortest diameters, the lengths of which are called the major and minor axes, respectively. The major axis, represented by \(2a\), runs through the center and both foci of the ellipse, ending at the widest part of the shape. The minor axis, represented by \(2b\), is perpendicular to the major axis, runs through the center, and ends at the narrowest part of the ellipse.

These axes not only define the overall shape and orientation of the ellipse but also provide vital information for calculating the area of the ellipse, which is given by \(\text{Area} = \text{π} \times a \times b\). It's crucial to know that all ellipses symmetrically divide along their axes, creating mirror-image halves. In any formula that involves an ellipse, the variable \(a\) will always stand for the semi-major axis and \(b\) for the semi-minor axis, ensuring the equation corresponds accurately to the ellipse's geometry.