Problem 100 - A Graphical Approach to College Algebra

Question

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 \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) as shown through distance simplification.

Step 1: Understanding the Distance Equation

The given equation \(|d(P, F') - d(P, F)| = 2a\) represents a hyperbola where \(F'\) and \(F\) are the foci. Here, \(P(x, y)\) is any point on the hyperbola.

Step 2: Distance from a Point to the Foci

The distance from a point \((x, y)\) to the foci \(F'( -c, 0)\) is \(\sqrt{(x + c)^2 + y^2}\) and to \(F(c, 0)\) is \(\sqrt{(x - c)^2 + y^2}\).

Step 3: Substitute Distances into Equation

Substitute the expressions for the distances into the hyperbola's equation: \[ |\sqrt{(x + c)^2 + y^2} - \sqrt{(x - c)^2 + y^2}| = 2a \]

Step 4: Square Both Sides

Square both sides to eliminate the absolute value and the nested square roots: \( ((x + c)^2 + y^2) - 2\sqrt{((x + c)^2 + y^2)((x - c)^2 + y^2)} + ((x - c)^2 + y^2) = (2a)^2 \).

Step 5: Simplify the Expressions

Simplify the expression: \( 2x^2 + 2y^2 + 2c^2 = 4a^2 + 2\sqrt{((x + c)^2 + y^2)((x - c)^2 + y^2)} \).

Step 6: Isolate the Square Root Term

Isolate the square root term: \(\sqrt{((x + c)^2 + y^2)((x - c)^2 + y^2)} = x^2 + y^2 + c^2 - 2a^2 \).

Step 7: Square Again to Simplify

Square both sides again: \(((x + c)^2 + y^2)((x - c)^2 + y^2) = (x^2 + y^2 + c^2 - 2a^2)^2 \).

Step 8: Introduce Variables

Use \(b^2 = c^2 - a^2\) and substitute into the expression, simplifying terms. Collect like terms to arrive at the standard form equation of a hyperbola.

Step 9: Derive the Hyperbola Equation

After substituting and simplifying, the equation reduces to: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), as required.

Key Concepts

Conic Sections

Conic sections are curves obtained from the intersection of a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas:

• **Circle**: All points equidistant from a center point.
• **Ellipse**: Sum of distances from two fixed points (foci) is constant.
• **Parabola**: Set of points equidistant from a focus and a directrix.
• **Hyperbola**: Difference of distances from two foci is constant.

Hyperbolas differ by having two separate curves, known as branches. The standard form equation of a hyperbola centered at the origin is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). The "minus" sign indicates its distinct nature among conics.

Foci of a Hyperbola

The foci are key points that define the hyperbola’s shape. For a hyperbola centered at the origin, the foci are located at \( F'(-c, 0) \) and \( F(c, 0) \).

• These points help determine the hyperbola's stretch along the x-axis.
• The constant \( 2a \) in the hyperbola's definition signifies the distance between branches, derived from the difference in distances to the foci.

Knowing the foci allows you to determine the relationship between \( a \), \( b \), and \( c \), where \( b^2 = c^2 - a^2 \). This relationship emerges when you equate and simplify the derived distances.

Distance Formula

The distance formula calculates the distance between two points in a plane. It is essential for understanding hyperbolas:

• The formula is \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
• For hyperbolas, it helps express the difference in distances from a point \( P(x, y) \) to the foci \( F'(-c, 0) \) and \( F(c, 0) \).

Using the formula, these distances can be written as \( \sqrt{(x + c)^2 + y^2} \) and \( \sqrt{(x - c)^2 + y^2} \). Squaring both sides of the equation \(|d(P, F') - d(P, F)| = 2a\) allows us to rearrange and simplify it to arrive at the hyperbola’s standard form equation.