Problem 5 - 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

The equation matches description E: Ellipse; center \((-2,4)\).

Step 1: Identify the Type of Conic Section

The given equation \(\frac{(x+2)^{2}}{9}+\frac{(y-4)^{2}}{25}=1\) has a form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). When both terms are added, and the coefficients are positive, it indicates this is an equation of an ellipse.

Step 2: Determine the Center of the Ellipse

The center \((h, k)\) of the ellipse is given in the equation, which can be identified as the opposite signs of the numbers alongside \(x\) and \(y\). Thus, the center is \((-2, 4)\).

Step 3: Check for the Description Match

Once we've determined that the figure is an ellipse centered at \((-2,4)\), we should look in Column II for the matching description. This description should be \(E.\) Ellipse; center \((-2,4)\).

Key Concepts

Ellipse

An ellipse is one of the four types of conic sections, formed by the intersection of a cone with a plane. The general equation of an ellipse in standard form is:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]This formula signifies a stretched circle, where:

• The point \(h, k\) is the center of the ellipse.
• The lengths \(a\) and \(b\) represent the distances from the center to the edge of the ellipse along the x-axis and y-axis respectively.

The larger value between \(a\) and \(b\) determines the direction of the ellipse's major axis. If \(a > b\), the ellipse is stretched along the x-axis. If \(b > a\), it is stretched along the y-axis. Ellipses are known for their symmetry and the property that the sum of the distances from any point on the ellipse to two given points, called foci, is constant. This distinct characteristic makes ellipses a special category of conic sections.
With real-world applications ranging from orbits of planets to architectural designs, ellipses play a crucial role in both mathematics and science.

Hyperbola

A hyperbola is another fascinating type of conic section that can be identified when a plane intersects both halves of a double cone. Its standard equation can be written as:\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]or\[ \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \]In these equations:

• The center of the hyperbola is at \(h, k\).
• The values \(a\) and \(b\) help define the hyperbola's shape and orientation.
• If the x-term is positive, the hyperbola opens horizontally, if the y-term is positive, it opens vertically.

A hyperbola consists of two symmetrical "open" curves. Unlike ellipses, the loci point diverge, making hyperbolas distinct as they get wider farther out. The unique feature of hyperbolas is that the difference of the distances from any point on the hyperbola to two fixed points (each called a focus) is a constant. Hyperbolas appear in various fields, illustrating phenomena such as certain reflective properties and the paths of objects under certain types of motion.

Center of a Conic Section

The center of a conic section is a pivotal point in understanding its geometry. It serves as the reference point from which the conic's principal axes extend. Each conic section - ellipse, circle, hyperbola, and parabola - uses this central point uniquely:

• For ellipses and hyperbolas, the center \(h, k\) is identified in their equations as the values affecting the terms \(x-h\) and \(y-k\).
• In the case of ellipses, the center is equidistant from the foci, situated at the midpoint of the line segment joining them.
• For hyperbolas, the center remains between its diverging branches and also equidistant from its vertices along the transverse axis.

In many real-world scenarios, correctly determining the center of a conic section is crucial for applying mathematical principles, especially in problems involving symmetry, range, and focal distances.
Understanding the center provides insight into the behavior and trajectory of the conic section, facilitating its more comprehensive analysis and application.