Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 33-36: Find an equation for the set of points in an \(x y\)-plane such that the sum of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$ F(3,0), \quad F^{\prime}(-3,0) ; \quad k=10 $$

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \).

Step 1: Identify the Type of Conic Section

Given that the sum of the distances from two fixed points (foci) \( F \) and \( F' \) to any other point on the curve is constant, we are dealing with an ellipse.

Step 2: Write the Ellipse Equation with Center at Origin

The general equation for an ellipse with foci along the x-axis is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where the semi-major axis \( a \) is half of the distance \( k \).

Step 3: Determine the Values of a and c

Given \( F = (3,0) \) and \( F' = (-3,0) \), the distance between the foci is \( 2c = 6 \), so \( c = 3 \). The constant sum \( k = 10 \), thus the semi-major axis \( a = \frac{10}{2} = 5 \).

Step 4: Calculate the Value of b

Using the relationship \( b^2 = a^2 - c^2 \), substitute \( a = 5 \) and \( c = 3 \):\[ b^2 = 5^2 - 3^2 = 25 - 9 = 16 \]So, \( b = 4 \).

Step 5: Write the Final Equation of the Ellipse

Now that we have \( a = 5 \) and \( b = 4 \), substitute these into the ellipse equation:\[ \frac{x^2}{5^2} + \frac{y^2}{4^2} = 1 \]Which simplifies to \[ \frac{x^2}{25} + \frac{y^2}{16} = 1 \].

Key Concepts

Understanding Ellipse

An ellipse is a fundamental concept in geometry and conic sections. It is essentially a stretched out circle. An ellipse evolves from two fixed points, known in geometry as the foci, which determine its shape. Any point on the boundary of the ellipse has a unique characteristic: the sum of its distances to the two foci is constant. This defining trait makes the ellipse unique among conic sections.

This constant sum condition is critical in forming the ellipse's shape and is denoted by the term "k" in equations. For instance, if you were to plot a series of points on paper using this distance rule, gradually you would connect enough points to visualize an oval-like structure, which is the ellipse itself. Think of how planets orbit the sun in elliptical shapes. Though different mechanics are at play, it's a real-world example of an ellipse.

Decoding the Distance Formula

The distance formula helps us calculate the distance between two points in a coordinate plane. It is vital in determining relationships between points, such as the foci in an ellipse.

The formula is as follows:

• For two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) is calculated using \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

When working with ellipses, the distance formula is pivotal for finding the distance between the center and a point on its boundary or between the foci themselves.

For instance, knowing the distance between foci or from foci to any ellipse point directly influences the equation parameters. In our problem, knowing the foci at \((3,0)\) and \((-3,0)\) helps verify the distances are used correctly later in determining the ellipse's dimensions.

Crafting the Ellipse Equation

The ellipse equation is a key tool for accurately describing the shape of an ellipse on a coordinate plane. It takes the general form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

This equation correlates with the ellipse's orientation and its axes dimensions:

• \(a\) is the semi-major axis, showing the ellipse's longest boundary stretch.
• \(b\) is the semi-minor axis, indicating the shorter span of the shape.

For our exercise, with a constant sum \(k = 10\), we find \(a\) by halving \(k\) to get \(a = 5\).

Next comes calculating \(b\) using the relation \(b^2 = a^2 - c^2\), where \(c\) is half the distance between the foci. This calculation ensures your ellipse correctly reflects its geometric properties. Hence in this scenario,\((3,0)\) and \((-3,0)\) place \(c = 3\), resulting in \(b = 4\). Substituting back, your final equation is \(\frac{x^2}{25} + \frac{y^2}{16} = 1\).

Pinpointing the Foci in Ellipses

Foci are crucial in understanding ellipses. These two distinct points, denoted by \(F\) and \(F'\), define an ellipse's structure and its equation's essential trait.

In an ellipse, remember:

• The sum of distances from any point on the ellipse boundary to each focus remains consistent as you travel around the shape.
• These two points determine the focal distance \((2c)\), where \(c\) is the distance from the ellipse's center to either focus.

With the given foci at \((3,0)\) and \((-3,0)\), make sure it conforms to \(c = 3\). The foci's positions directly influence the stretch seen in an ellipse, guiding accurate shape depiction. They are essential in genuine astronomical and engineering applications as they relate to paths orbiting structures or forces directed along such paths.