Problem 19 - A Graphical Approach to Precalculus with Limits

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$4(x-3)^{2}+3(y+4)^{2}=0$$

Step-by-Step Solution

Answer

The "graph" of the equation is a single point at (3, -4), not a conic section.

Step 1: Understanding the Given Equation

The given equation is \(4(x-3)^2 + 3(y+4)^2 = 0\). This resembles the standard form of an ellipse, \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), or a circle, when both terms are positive and equal.

Step 2: Analyzing the Equation Structure

Observe that the sum of two squared terms \(4(x-3)^2\) and \(3(y+4)^2\) is set equal to zero. In the standard equations for conic sections (ellipse, hyperbola, circle, and parabola), the right side is usually a positive number.

Step 3: Checking the Equation for a Graph

Since \(4(x-3)^2 + 3(y+4)^2 = 0\) is a sum of non-negative terms (squares), and squares cannot produce a negative total unless both are zero, this equation implies both terms must individually be zero: \((x-3)^2 = 0\) and \((y+4)^2 = 0\).

Step 4: Implicit Solution to Individual Terms

Solving \((x-3)^2 = 0\) gives \(x = 3\), and solving \((y+4)^2 = 0\) gives \(y = -4\). The equation only has a solution at the point \((3, -4)\).

Step 5: Conclusion on the Type of Graph

Since the equation does not satisfy the standard form of any conic section for a range of values (but instead defines a single point), the graph is not that of a conic section.

Key Concepts

Equations of Conic Sections

Conic sections, or conics, are the curves obtained by intersecting a plane with a double-napped cone. The primary types of conics are circles, ellipses, parabolas, and hyperbolas. Each of these types has a standard equation in either Cartesian or polar coordinates. To identify the graph derived from a conic section equation, it is essential to recognize the specific form it takes. For instance, the equation of an ellipse typically looks like this: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).

• If \( a = b \), the ellipse becomes a circle.
• For a hyperbola, the equation is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).
• A parabola has the form \( y = ax^2 + bx + c \).

While these standard forms provide a clear idea of each conic, identifying conics solely through equations can often be challenging. Simplifying or rearranging the terms typically aids in discerning their type.

Ellipses

An ellipse might look complicated at first glance, but it's quite straightforward! Ellipses are the "stretched" circles you see around. They have two focal points (foci) and exhibit symmetry around both the x- and y-axes. The standard equation for ellipses is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \( (h, k) \) is the center of the ellipse.If you look closely:

• \(a^2\) is always under \((x-h)^2\) if the ellipse is wider horizontally.
• \(b^2\) is under \((y-k)^2\) if the ellipse is taller vertically.

The value under each squared term tells you how "stretched" the ellipse is in that direction. The larger the denominator, the more stretched it becomes. Learning to identify and manipulate these equations helps in solving geometric problems.

Hyperbolas

Hyperbolas look like two curves waving away from each other. They might seem unusual, but they're fascinating to study. Their standard form equation is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \). Using this, you can determine their orientation:

• When the \(x\)-term (\((x-h)^2/a^2\)) is positive, the hyperbola opens horizontally.
• If the \(y\)-term (\((y-k)^2/b^2\)) is positive, it opens vertically.

Hyperbolas have two distinct branches, and each branch is reflective over a central point called the center. This center is located at \((h, k)\). The values \(a\) and \(b\) help determine the slope of the asymptotes and the shape of the branches. Understanding these details can simplify graphing these intriguing curves.

Circles

Circles are usually the first conic section students learn about, given their simple and symmetrical nature. They are perfect forms and extremely easy to work with. The standard equation for a circle is depicted as \( (x-h)^2 + (y-k)^2 = r^2 \). Here,

• \((h, k)\) represents the circle's center.
• \(r\) is the radius of the circle.

In this equation, any changes to \(h\) and \(k\) simply move the circle's center without altering its size. Adjusting \(r\), however, changes the size, stretching or shrinking the circle. When analyzing a conic equation, if both squared terms are present and their coefficients are equal, then you're looking at a circle! Circles are straightforward geometrically and algebraically, making them accessible and essential early learning.