Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. \(\mathbf{II}\) 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 describes an ellipse with a vertical major axis and center at \((2, -4)\), matching description H.

Step 1: Identify the Type of Equation

The given equation \(\frac{(x-2)^{2}}{9}+\frac{(y+4)^{2}}{25}=1\) is in the standard form of an ellipse because the sum is equal to 1 (ellipses and circles have the equal-to-1 form, but this is not a circle due to differing denominator values). An ellipse in standard form is \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), where \((h, k)\) is the center.

Step 2: Determine the Center of the Ellipse

Comparing with the standard form \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), we identify that the center \((h, k)\) of the ellipse is \((2, -4)\). Thus, the center matches with description H (Vertical major axis; center \((2,-4)\)).

Step 3: Determine the Major Axis

To determine the orientation of the ellipse, compare \(a^{2}\) and \(b^{2}\). Here, \(a^{2} = 9\) and \(b^{2} = 25\). The major axis is along the \(y\)-axis since \(b^{2} > a^{2}\), which is the criterion for having a vertical major axis.

Key Concepts

Ellipse

An ellipse can be thought of as an elongated circle, representing all points where the sum of the distances to two fixed points (the foci) is constant. Unlike circles, ellipses have two axes of symmetry: the major axis, which is the longest diameter, and the minor axis, which is shorter. You can describe an ellipse by this key equation:

• \[ \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \]

Here,

• x and y are the coordinates of any point on the ellipse,
• h and k are the x and y coordinates of the center, respectively,
• a and b are the distances from the center to the vertices along the major and minor axes.

If a equals b, you have a circle instead of an ellipse. In ellipses, the denominators are crucial as they determine the lengths of the axes. Larger denominators suggest longer axes.

Center of Ellipse

The center of an ellipse is a crucial reference point from which measurements along the axes are taken. It's the midpoint between its foci. In the equation \[\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\], the center is defined by the coordinates \((h, k)\).
For example, a given ellipse equation like \[\frac{(x-2)^{2}}{9} + \frac{(y+4)^{2}}{25} = 1\], has a center at \((2, -4)\).
By identifying this, you determine the starting point for locating the axes and potentially the orientation of the ellipse itself. Knowing the center is also fundamental for graphing the ellipse accurately. It can help understand the symmetry and overall shape of the ellipse.

Major Axis Orientation

The orientation of the major axis of an ellipse tells you which direction the longest part of the ellipse stretches. The formula

• \(a^{2} > b^{2}\) indicates a horizontal major axis,
• while \(b^{2} > a^{2}\) indicates a vertical major axis.

In an equation like\[\frac{(x-2)^{2}}{9} + \frac{(y+4)^{2}}{25} = 1\],you observe that \(b^{2} = 25\) (larger value) compared to \(a^{2} = 9\).
This results in a vertical major axis. This orientation helps predict how the ellipse is stretched relative to its center and is crucial for graphing it. Always compare the values of \(a^{2}\) and \(b^{2}\) to discern the orientation quickly and correctly.