Problem 44 - Precalculus: Mathematics for Calculus

Question

Find an equation for the conic section with the given properties. The ellipse with foci \(F_{1}(3,-4)\) and \(F_{2}(3,4),\) and \(x\) -intercepts 0 and 6

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{(x - 3)^2}{9} + \frac{y^2}{25} = 1 \).

Step 1: Find midpoint between foci

The foci of the ellipse are given as \(F_1(3,-4)\) and \(F_2(3,4)\). To find the center of the ellipse, we calculate the midpoint of these two points:\[\left( \frac{3 + 3}{2}, \frac{-4 + 4}{2} \right) = (3, 0)\]So, the center of the ellipse is \((3, 0)\).

Step 2: Determine the orientation of the ellipse

Since the foci have the same x-coordinate, and they differ in the y-coordinate, the major axis of the ellipse is vertical.

Step 3: Calculate the distance between the center and each focus

The distance between the center \((3, 0)\) and each focus \((3, 4)\) or \((3, -4)\) is 4. This distance is the value of \(c\), which is the distance from the center to each focus:\[c = 4\]

Step 4: Use x-intercepts to find semi-major and semi-minor axes

The given \(x\)-intercepts are 0 and 6, which means the ellipse crosses the x-axis at these points. The distance between these intercepts is 6, which is twice the semi-minor axis length \(2b\):\[b = 3\]

Step 5: Use the relationship between a, b, and c to find a

We know the relationship between the semi-major axis \(a\), semi-minor axis \(b\), and \(c\) is:\[a^2 = b^2 + c^2\]Substitute \(b = 3\) and \(c = 4\):\[a^2 = 3^2 + 4^2 = 9 + 16 = 25\]\[a = 5\]

Step 6: Write the equation of the ellipse

The center of the ellipse is \((3, 0)\). Since the major axis is vertical, the equation of the ellipse is:\[\frac{(x - 3)^2}{b^2} + \frac{(y - 0)^2}{a^2} = 1\]Substitute \(b = 3\) and \(a = 5\):\[\frac{(x - 3)^2}{9} + \frac{y^2}{25} = 1\]

Key Concepts

Ellipses

An ellipse is a fascinating shape and is one of the primary conic sections. Imagine squishing a circle along one of its diameters. The result is an elongated curve named a "circle with two centers," or simply an ellipse. This shape has two significant parts: the major and minor axes. The major axis is the longest diameter that runs through the center and both foci. The minor axis is perpendicular to the major axis and passes through the center but not the foci. Each of these axes is pivotal in determining the shape of the ellipse.

• **Major Axis:** Longer diameter containing both foci.
• **Minor Axis:** Shorter diameter that is perpendicular to the major axis.

An ellipse has two fixed points called foci (singular: focus). The sum of the distances from any point on the ellipse to these foci is constant. Compared to a circle, where the distance from the center to any point on the circumference is always the same, the ellipse changes in one direction but maintains a balance thanks to its properties.

Foci of Ellipse

The foci of an ellipse are two distinct points located along the major axis. They are crucial in defining the shape of an ellipse. In the given exercise, the foci are positioned at

• **Foci:** - \( F_1(3, -4) \) - \( F_2(3, 4) \)

These points are particularly important because they help determine the eccentricity of an ellipse. The eccentricity (often denoted as \( e \)) gives us an indication of how "stretched" an ellipse is. It is calculated using the formula \( e = \frac{c}{a} \), where \( c \) is the distance from the center to a focus, and \( a \) is the semi-major axis.
The fundamental property of an ellipse is that the total distance from any point on the ellipse to both foci is constant. In other words, for any point \( P \) on the ellipse, the sum of distances \( PF_1 + PF_2 \) remains the same.

Equation of Ellipse

The equation of an ellipse can express the perfect balance between its axes and foci. This mathematical formula allows you to describe the shape and orientation of an ellipse.
In the standard form, an ellipse equation centered at - \((h, k)\) is written as:i\[\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1\]where:

• \(a\) is the semi-major axis length
• \(b\) is the semi-minor axis length
• \((h, k)\) is the center of the ellipse

For example, in the given problem with a center at \( (3, 0) \), the lengths of \(a = 5\) and \(b = 3\) allow us to craft the equation as \[\frac{(x - 3)^2}{9} + \frac{y^2}{25} = 1\]. The equation shows that the ellipse is elongated along the y-axis. This complete system signifies an impeccable combination of geometry and algebra, modeling the shape precisely.