Problem 8 - A Graphical Approach to College Algebra

Question

Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. A. Hyperbola; center \((2,4)\) B. Ellipse; foci \((\pm 2 \sqrt{3}, 0)\) C. Hyperbola; foci \((0, \pm 2 \sqrt{5})\) D. Hyperbola; center \((-2,4)\) E. Ellipse; center \((-2,4)\) F. Center \((0,0) ;\) horizontal transverse axis G. Ellipse; foci \((0, \pm 2 \sqrt{3})\) H. Vertical major axis; center \((2,-4)\) $$\frac{(x-2)^{2}}{9}-\frac{(y-4)^{2}}{25}=1$$

Step-by-Step Solution

Answer

A: Hyperbola; center \((2,4)\).

Step 1: Identify the Conic Section

The given equation is \( \frac{(x-2)^{2}}{9} - \frac{(y-4)^{2}}{25} = 1 \). The standard form of a hyperbola is \( \frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1 \) or \( \frac{(y-k)^{2}}{a^{2}} - \frac{(x-h)^{2}}{b^{2}} = 1 \). Since the equation has the subtraction sign in between, it is a hyperbola.

Step 2: Determine the Center

The center \((h, k)\) of the hyperbola can be identified from the equation \( \frac{(x-2)^{2}}{9} - \frac{(y-4)^{2}}{25} = 1 \) as \((2, 4)\).

Step 3: Confirm Orientation

For the given hyperbola \( \frac{(x-2)^{2}}{9} - \frac{(y-4)^{2}}{25} = 1 \), as the \(x\) term is positive, this hyperbola opens left and right, confirming a horizontal transverse axis.

Step 4: Match with Given Descriptions

The hyperbola has a center at \((2, 4)\), and it opens horizontally, as calculated. This matches description A: Hyperbola; center \((2,4)\).

Key Concepts

Hyperbola

Hyperbolas are fascinating and highly useful shapes found within the broader study of conic sections. A hyperbola is defined as the set of all points in the plane where the absolute difference of the distances from two foci is constant. This distinct path creates a set of curves that open outward, away from each other, in two distinct branches.
These curves are often seen in various real-world applications, such as in navigation and radio signals, where the properties of hyperbolas have practical uses.A hyperbola contains two axes:

• The transverse axis – the line segment connecting the vertices of the hyperbola, determined by the positioning and orientation of the hyperbola.


• The conjugate axis – perpendicular to the transverse axis and helps in defining the complete shape.

In a Cartesian plane, the standard form equation for a hyperbola is either:

• \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] – here, the transverse axis is horizontal.


• \[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \] – here, the transverse axis is vertical.

The subtraction sign in the equation indicates that the conic is indeed a hyperbola. Recognizing the direction and distinctive features, such as branch orientation, are key elements when working with hyperbolas.

Center of Hyperbola

The center of a hyperbola is a significant starting point that guides the formation of the hyperbola's curves. This center is mathematically denoted as \(h, k\) and can be identified from the standard form of the hyperbola equation. By analyzing where the equation's variables are shifted, one can pinpoint this essential center.
For example, in the equation \[ \frac{(x-2)^2}{9} - \frac{(y-4)^2}{25} = 1 \], the center is located at \(2, 4\). These coordinates act as a balance point from which the hyperbola opens symmetrically in opposite directions.To determine the center:

• Identify the constants \(h\) and \(k\) in the equation \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).


• These values represent horizontal and vertical shifts from the origin (0,0).


• The center guides the intersection of the transverse and conjugate axes.

The center isn't just a mere location; it plays a critical role in understanding and plotting every other part of a hyperbola. Knowing the center, helps in sketching and navigating the hyperbola on a coordinate plane.

Equation of Conic Sections

Conic sections comprise important geometric shapes, including circles, ellipses, parabolas, and hyperbolas. Each type has its characteristic equation, determined by slicing cones at varying angles. These equations not only define the shapes but also unlock a world of analytical geometry useful in multiple fields like astronomy and engineering.
Common characteristics of conic section equations:

• Circles: The simplest form, where the distance from the center to any point on the perimeter is constant, given by \((x-h)^2 + (y-k)^2 = r^2\).


• Ellipses: Defined by the sum of distances to two foci being constant, represented by \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \].


• Parabolas: Characterized by a single focal point, leading to the equation form \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\).


• Hyperbolas: As previously conveyed, recognizable by the equation \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] with distinguishing subtraction.

Each conic tells a unique story of geometry and algebra intersecting, where manipulation of their equations helps in identifying physical and theoretical properties of the curves they represent. Whether predicting planetary motion or designing reflective properties in optics, understanding the equations of conic sections provides a pivotal advantage.