Problem 44 - Precalculus: Functions and Graphs

Question

Exer. \(41-44:\) Find an equation for the set of points in an Xy-plane such that the sum of the distances from \(F\) and \(F\) is \(k\) $$F(0,8), \quad F(0,-8) ; \quad k=20$$

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{100} + \frac{y^2}{164} = 1\).

Step 1: Identify Relevance to an Ellipse

The problem given describes a situation where the sum of the distances from two fixed points (foci) is constant for all points on the curve. This is the definition of an ellipse.

Step 2: Identify the Coordinates of Foci

The coordinates of the foci provided are \((0,8)\) and \((0,-8)\). Given these coordinates, the center of the ellipse is at the origin \((0,0)\).

Step 3: Determine the Distance Between Foci

Calculate the distance between the foci using the formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Here, it simplifies to \(\sqrt{(0-0)^2 + (-8-8)^2} = \sqrt{256} = 16\).

Step 4: Set Up the Ellipse Equation

For an ellipse with foci \((0,c)\) and \((0,-c)\), the equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(2a = k\) and \(c^2 = b^2 - a^2\).

Step 5: Calculate Parameters: a and b

\(2a = 20\), therefore \(a = 10\). The foci are \(\pm c = \pm 8\). Use \(c^2 = b^2 - a^2\) to find \(b\): \(c = 8\), \(b^2 = a^2 + c^2 = 10^2 + 8^2 = 100 + 64 = 164\).Therefore, \(b = \sqrt{164}\).

Step 6: Write the Equation of the Ellipse

Substitute the values of \(a\) and \(b\) into the ellipse equation: \[ \frac{x^2}{100} + \frac{y^2}{164} = 1 \] This is the equation of the ellipse.

Key Concepts

Foci of an Ellipse

An ellipse is a unique shape in geometry that has two specific points called foci (singular: focus). These foci are not just important, they are fundamental in defining what an ellipse is. For any point on the ellipse, the sum of the distances from that point to each focus remains constant. In the exercise given, the foci are located at coordinates \((0, 8)\) and \((0, -8)\). Here, both foci are aligned vertically along the y-axis.
Typically, foci help determine the shape and orientation of the ellipse. If they are wide apart, the ellipse appears more stretched. If they are closer, the ellipse appears more circular. Understanding the role of foci helps visualize how the ellipse elongates in one direction more than the other, which is key in comprehending its geometric properties.

Distance Formula

The distance formula is a crucial tool for calculating the distance between two points in a plane. This formula, \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), helps find the straight-line distance regardless of where the points are situated on the coordinate plane. In our exercise, the distance between the foci, \((0,8)\) and \((0,-8)\), is calculated.
Calculating this distance is important because it directly influences the parameters of the ellipse. Here, the distance was found to be 16 units which is a key part of setting up the ellipse equation. By mastering how to use the distance formula, students can easily transition into understanding more complex geometric shapes and their properties.

Ellipse Parameters

When working with ellipses, there are specific parameters that describe their size and shape: \(a\), \(b\), and \(c\).
* \(a\) represents the semi-major axis, which is half of the longest diameter of the ellipse.
* \(b\) represents the semi-minor axis that is half of the shortest diameter.
* \(c\) is the distance from the center of the ellipse to each focus.
In the exercise, given that \(2a = 20\), it follows that \(a = 10\). The parameters are crucial because they are necessary for constructing the ellipse equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Further, the parameter \(c\) relates to \(a\) and \(b\) through the equation \(c^2 = b^2 - a^2\). For this ellipse, with foci at \(c = 8\), it was calculated that \(b = \sqrt{164}\). Together, these parameters define the precise nature and orientation of the ellipse.

XY Plane Geometry

Geometry within the XY plane involves analyzing shapes, lines, and curves plotted on a two-dimensional plane. Understanding geometry in this plane is fundamental for many mathematical applications including calculus, physics, and engineering.
In the given exercise, identifying the ellipse in the XY plane involves determining how the curve relates to the x and y axes, which are foundational lines in this geometry.
By comprehending how to construct and interpret an ellipse's equation \(\frac{x^2}{100} + \frac{y^2}{164} = 1\), students can see how the shape fits into this plane. It helps them understand its dimensions, how it spreads across the axes, and where its foci lie in relation to the center at \((0,0)\). Mastering XY plane geometry is integral to visualizing and solving complex geometric problems.