Problem 100 - A Graphical Approach to Precalculus with Limits

Question

Equation of a Hyperbola Suppose a hyperbola has center at the origin, foci at \(F^{\prime}(-c, 0)\) and \(F(c, 0),\) and equation $$ \left|d\left(P, F^{\prime}\right)-d(P, F)\right|=2 a $$ Let \(b^{2}=c^{2}-a^{2},\) and show that the points on the hyperbola satisfy the equation $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $$

Step-by-Step Solution

Answer

The points on the hyperbola satisfy \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 1: Understand Hyperbola Definition

A hyperbola is defined as the set of points \((x, y)\) such that the absolute difference of distances from two foci, \(F'(-c,0)\) and \(F(c,0)\), is constant, \(2a\). This translates the definition of a hyperbola from a geometric to an algebraic condition.

Step 2: Use Distance Formula

The distance from a point \((x, y)\) to a focus \(F'(-c, 0)\) is given by \(d(P, F') = \sqrt{(x+c)^2 + y^2}\) and to the other focus \(F(c, 0)\) by \(d(P, F) = \sqrt{(x-c)^2 + y^2}\).

Step 3: Apply Hyperbola Condition

We apply the condition of the hyperbola: \( \left|d(P, F') - d(P, F)\right| = 2a \). Substituting the distances calculated, we arrive at the equation: \[ \left|\sqrt{(x+c)^2 + y^2} - \sqrt{(x-c)^2+y^2}\right| = 2a \].

Step 4: Simplify and Square the Equation

To remove the square roots, square both sides of the equation. This gives \((\sqrt{(x+c)^2 + y^2} - \sqrt{(x-c)^2+y^2})^2 = (2a)^2\). Simplify to derive the equation in terms of \((x, y)\),\((x^2/c^2)-(y^2/b^2)=1\).

Step 5: Derive Hyperbola Equation

Recognize that the hyperbola's standard form is \(x^2/a^2 - y^2/b^2 = 1\). Recall the relationship \(b^2 = c^2 - a^2\), then substitute to show that the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) follows.

Key Concepts

Distance Formula and Its Application in Hyperbolas

The distance formula is crucial for understanding the structure of a hyperbola. It helps calculate the distance between two points in a plane. Given a point \(x, y\), and a focus \(F(c, 0)\), the distance formula is expressed as \(d(P, F) = \sqrt{(x-c)^2 + y^2}\). Application of this formula is key to solving hyperbolic equations.
To start, calculate the distances from any point \(P(x, y)\) on a hyperbola to each focus point. Using the focus \(F'(-c, 0)\), the formula becomes \(d(P, F') = \sqrt{(x+c)^2 + y^2}\). These calculations will be essential for manipulating the equation of the hyperbola into its standard form. By understanding the role of these distances, you can faithfully convert the geometric definition of a hyperbola into an algebraic equation.

Geometric Definition of Hyperbolas

A hyperbola is a set of all points \((x, y)\) such that the absolute difference of the distances from two fixed points, called the foci, is constant. This condition delineates the unique shape of the hyperbola, which consists of two symmetrical curves. The centers of these curves are at the origin \((0,0)\) when the foci \(F'(-c,0)\) and \(F(c,0)\) are symmetrically placed along a horizontal line.
To visualize this, think of the hyperbola as similar to an hourglass, open on two opposite sides. The distance between the branches increases as you move away from the center. For any point \(P(x, y)\) on this curve, the measured distances to each focus \(F \text{ and } F'\) differ by a constant value, \(2a\). Understanding this property is key to grasping the concept and rules governing hyperbolas.

Deriving the Algebraic Equation of a Hyperbola

The equation of a hyperbola bridges the geometric intuition with algebraic expression, facilitating calculations and problem-solving. To begin, consider the geometric definition where the absolute difference between distances to the foci meets the condition \( |d(P, F') - d(P, F)| = 2a\). Solving for the algebraic equation involves using the distance formula for both foci and taking the absolute difference leading to a simplified form.
Substituting the distance expressions, \(d(P, F') = \sqrt{(x+c)^2 + y^2}\) and \(d(P, F) = \sqrt{(x-c)^2 + y^2}\), into the hyperbola condition helps us derive the equation with care. Squaring both sides, another algebraic step, is crucial for turning the equation into a standard form. The algebraic process highlights the transition from geometric to algebraic thinking.

Standard Form of Conic Sections

Conic sections, including hyperbolas, have specific standard forms that simplify their analysis and categorization. The standard form of a hyperbola is given by \((x^2/a^2) - (y^2/b^2) = 1\).
This form allows for easy manipulation and graphing of the hyperbola to better understand its properties. Here, \(a\) represents the distance from the center to the vertices along the transverse axis, and \(b\) is linked with the slope of the asymptotes, related by \(b^2 = c^2 - a^2\). Understanding these relations is critical in placing a hyperbola correctly on a coordinate system, predicting its curve, and analyzing its physical properties, such as asymptotes and orientation.