Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(4 x^{2}+25 y^{2}+16 x+250 y+541=0\)

Step-by-Step Solution

Answer

The graph of the given equation is an ellipse.

Step 1: Rewrite the Equation into Standard Form

Rewrite the equation into the standard form for each type of conic section. Here, we can group the \(x\) terms and the \(y\) terms: \(4 x^{2}+16 x+25 y^{2}+250 y+541=0\). This can be rewritten as: \(4(x^{2}+4x) + 25 (y^{2} + 10y) = -541\). Completing the square for each variable, we balance the equation on each side and find the values of \(h\) and \(k\). By completing the square: \(4[(x+2)^{2}-4] + 25[(y+5)^{2}-25] = -541\). Expanding and simplifying we obtain: \(4(x+2)^{2} + 25(y+5)^{2}= 441\). Dividing by 441 to normalize the equation, we get \(\frac{(x+2)^{2}}{110.25} + \frac{(y+5)^{2}}{17.64} = 1\). Comparing this equation to the standard form of an ellipse, we see that they match.

Step 2: Identify the Conic Section

The equation \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\) is the standard form for an ellipse, where \((h, k)\) is the center of the ellipse and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. Thus, the given equation represents an ellipse.

Key Concepts

Ellipse

An ellipse is a significant shape in the study of conic sections. It resembles an elongated circle and possesses unique properties that distinguish it from other conic sections like parabolas or hyperbolas. An ellipse is defined as the collection of points such that the sum of the distances from any point on the ellipse to two fixed points (called foci) is constant.

Here are some basic characteristics of an ellipse:

• The center of the ellipse is the midpoint between the two foci.
• It has two axes of symmetry; the longest axis is the major axis, and the shortest is the minor axis.
• The ellipse is symmetrical along both axes.

The understanding of ellipses is vital because they occur naturally in physics, particularly in the orbits of celestial bodies such as planets and satellites.

Completing the Square

Completing the square is a useful algebraic technique often used to transform a quadratic expression into a perfect square trinomial. This process is crucial when rewriting a conic section equation into its standard form. By completing the square, you can reveal important characteristics of the conic, such as the center and radii of an ellipse.

The basic process of completing the square for a quadratic \(ax^2+bx+c\) involves forming \((x-h)^2 = 0\) by determining \(h\) from \(b\):

• Factor out \(a\) from the x-related terms.
• Take half of the coefficient of \(x\) (i.e., \(\frac{b}{2a}\)), square it, and add and subtract it within the equation.
• Reorganize the equation to isolate the perfect square trinomial.

This technique was demonstrated effectively in the given problem when rewriting the original equation to standard form by adjusting both \(x\) and \(y\) terms.

Standard Form of an Ellipse

The standard form of an ellipse equation is crucial for understanding the ellipse's properties and graphing it accurately. An ellipse in the standard form is represented as: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse.

Key elements of the standard form include:

• \(a\), representing the semi-major axis, is always equal to or larger than \(b\).
• \(b\), representing the semi-minor axis.
• The lengths \(2a\) and \(2b\) provide the total widths of the ellipse through its major and minor axes, respectively.

Using this standard form allows for a detailed understanding of the ellipse's layout and geometry. Moreover, it's easy to calculate the ellipse's foci and verify its position and orientation when rearranged correctly using algebraic methods like completing the square.