Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(13,0), \quad F^{\prime}(-13,0) ; \quad k=24$$

Step-by-Step Solution

Answer

The equation is \( \frac{x^2}{144} - \frac{y^2}{25} = 1 \).

Step 1: Understanding the Problem

The goal is to find the equation representing a set of points where the difference in distances from two fixed points, \( F(13,0) \) and \( F'(-13,0) \), is equal to \( k \). This describes a hyperbola.

Step 2: Equation of a Hyperbola

The general equation for a hyperbola centered at the origin with foci along the x-axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). The distance between the foci is identified as \( 2c \) in terms of \( c = \sqrt{a^2 + b^2} \).

Step 3: Identifying Parameters

Given points \( F(13,0) \) and \( F'(-13,0) \), we have \( 2c = 26 \) thus \( c = 13 \). The given \( k \), the difference of distances to the foci, equals \( 2a = 24 \), so \( a = 12 \).

Step 4: Relate and Solve for \( b^2 \)

Use \( c^2 = a^2 + b^2 \):\[13^2 = 12^2 + b^2 \rightarrow 169 = 144 + b^2 \rightarrow b^2 = 25\].

Step 5: Write the Hyperbola Equation

Now, substitute \( a^2 \), \( b^2 \) into the hyperbola equation:\[\frac{x^2}{144} - \frac{y^2}{25} = 1\].

Key Concepts

Distance Formula

In mathematics, the distance formula is a crucial tool used to calculate the distance between two points in the coordinate plane. For points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is given by the formula:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

This formula is derived from the Pythagorean theorem. It essentially forms a right-angled triangle with the line connecting the two points as the hypotenuse.
Understanding this formula is crucial for exploring hyperbolas and other conic sections, as these involve comparisons of distances from certain points to define their unique shapes.

Foci of a Hyperbola

Hyperbolas are defined by their foci, which are two fixed points located symmetrically on either side of the hyperbola’s center. For any point on the hyperbola, the absolute difference in distances to these two foci is a constant, known in problems as \(k\).
The foci are critical when constructing the equation of a hyperbola. For the problem at hand, the foci are given as \(F(13,0)\) and \(F'(-13,0)\), meaning they lie symmetrically along the x-axis from the origin.

• Distance between the foci is represented as \(2c\).
• Here, \(c = 13\) as calculated from \(2c = 26\).

The foci help determine other parameters required for the hyperbola's equation, making them indispensable in studying conic sections.

Conic Sections

Conic sections refer to the shapes obtained by intersecting a plane with a cone. The primary types include circles, ellipses, parabolas, and hyperbolas. Each has distinct properties and equations that dictate its form and behavior.
A hyperbola, one of these conic sections, is defined by its 'two-part' nature, often appearing as two mirror-image curves opening away from each other.

• They are characterized by being the locus of points for which the absolute difference in distances to two foci is constant.
• These properties lead to the unique hyperbola equation form.

Understanding conic sections is fundamental to exploring advanced topics in geometry and physics, particularly in analyzing trajectories and orbital paths.

Equation of a Hyperbola

The equation of a hyperbola is an algebraic representation of its geometric properties. For a hyperbola centered at the origin, with its foci along the x-axis, the standard form of the equation is:

• \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

Here, \(a\) is half of the distance between the vertices, and \(b\) is associated with the semi-minor axis, which can be calculated using \(c^2 = a^2 + b^2\), where \(c\) is half the distance between the foci.
In our given problem, with \(a = 12\) determined from \(2a = 24\) and \(c = 13\), we find \(b^2 = 25\) using the relation \(c^2 = a^2 + b^2\).
Finally, we substitute these values into the standard form to get:

• \(\frac{x^2}{144} - \frac{y^2}{25} = 1\)

This equation concisely describes the hyperbola's structure in the coordinate plane, linking its geometric layout to algebraic expression.