Problem 52 - Algebra and Trigonometry

Question

In Exercises 49–56, identify each equation without completing the square. $$ 9 x^{2}+25 y^{2}-54 x-200 y+256=0 $$

Step-by-Step Solution

Answer

The equation \(9x^2 + 25y^2 -54x -200y + 256 = 0\) is the equation of an ellipse.

Step 1: Group the x and y terms together

We rearrange the equation to group the x and y terms together as follows: \(9(x^2 - 6x) + 25(y^2 - 8y) = -256\)

Step 2: Identify the form

We recognize that it is the equation of an ellipse in the form \(A(x-h)^2 + B(y-k)^2 = 1\).

Key Concepts

Completing the Square

Completing the square is a method used to transform quadratic equations into a perfect square trinomial. This technique is often used to rewrite conic sections, including ellipses, into their standard forms.
To better understand this process, let's take a closer look at how you can complete the square for the x terms in an equation. Consider a quadratic expression like \(ax^2 + bx\). First, you need to ensure the coefficient of \(x^2\) is 1 by factoring out a if necessary.

• Next, take half of the coefficient of the \(x\) term, square it, and add and subtract it inside the parentheses. This creates a perfect square trinomial within the equation.
• For example, with the term \(x^2 - 6x\), half of \(-6\) is \(-3\), and squaring it gives \(9\). Thus, we rewrite it as \((x-3)^2 - 9\).

By completing the square for both x and y terms, you turn the original equation into its simplified, standard form.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These sections can take different shapes, such as circles, ellipses, parabolas, and hyperbolas. These shapes result from different angles or positions at which the plane intersects the cone. Understanding conic sections is essential in geometry and helps to analyze and interpret various properties of these curves.
An ellipse, one specific form of a conic section, is significant when dealing with equations like the one in our exercise. If the plane cuts through the cone at an angle, forming a closed curve, this results in an ellipse.

• The general equation for a conic section is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
• Depending on the values of the coefficients, the equation can be manipulated to represent a specific conic section, such as an ellipse, which is the case here when there are no B and D terms, and A ≠ C.

Understanding this allows you to determine the type of conic section an equation represents by analyzing its structure and transforming it accordingly.

Standard Form of an Ellipse

The standard form of an ellipse helps in understanding the fundamental properties of the shape, such as its center and radii, which describe the ellipse's size and position.
The equation in the standard form for an ellipse with a horizontal major axis is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
Here, \((h, k)\) represents the center of the ellipse, \(a\) is the semi-major axis' length, and \(b\) is the semi-minor axis' length.

• To convert an equation into this standard form, like the one given in the exercise, you would first use completing the square to organize the equation.
• This reorganization allows you to clearly identify \((h, k)\), \(a\), and \(b\), providing a concise description of the ellipse's geometry.

The standard form is crucial, as it simplifies the analysis of the ellipse and its graphical representation, making it easier to interpret and work with in various applications.