Problem 67 - Precalculus with Limits

Question

In Exercises 57-72, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(25y^2-10x-200y-119=0\)

Step-by-Step Solution

Answer

The graph of the given equation represents a parabola.

Step 1: Group the Terms

Group the terms that contain the same variable together: \(25y^2 - 200y - 10x = 119\).

Step 2: Complete the Square for y

To complete the square for y, the coefficient before \(y^2\) must be 1. Thus, factor out 25 from the y terms, then add the square of half the coefficient of y inside the brackets: \(25(y^2 - 8y + 16) - 10x -119 - 25*16 = 0\)This simplifies to: \( 25(y - 4)^2 - 10x - 519 = 0\).

Step 3: Identify the Type of Graph

The equation \(25(y - 4)^2 - 10x - 519 = 0\) is not yet in the standard form for a circle, parabola, ellipse, or a hyperbola. However, it has the form \(ax^2 - by = c\) which is a parabola opening either to the left or right depending on the sign of x (negative x means the parabola opens to the left), given that a, b and c are constants.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations, convert the equation of a parabola to its standard form, or find the vertex of the parabola. It involves rewriting a quadratic expression into a perfect square trinomial plus a constant.

For example, for a quadratic expression in the form of \( ax^2 + bx + c \), the goal is to create a perfect square from the terms containing the variable. Let's delve deeper into the exercise-related process. The coefficient of the squared term should be one for the process to work smoothly. If not, as in our case, we factor out the coefficient of \( y^2 \), here being 25, to get \( 25(y^2 - 8y + 16) \), which sets the stage for completing the square.

The next step is to determine what number needs to be added and subtracted to form a perfect square trinomial. This number is found by taking half of the linear coefficient \( -8 \) in our case, squaring it, and then adding it inside the parentheses, resulting in \( 16 \). The equation now becomes \( 25(y - 4)^2 \) after adding and subtracting \( 25 \times 16 \), balancing the equation. Thus, completing the square helps to transform the quadratic equation into a more workable form for further analysis.

Parabola Classification

Parabola classification is essential to understand the orientation and the shape of the graph produced by a quadratic equation. A parabola is one of the conic sections and it generally represents the graph of a quadratic function. The standard form of a quadratic equation is \( y = ax^2 + bx + c \) for parabolas that open up or down, or \( x = ay^2 + by + c \) for parabolas that open to the left or to the right.

In our exercise solution, after completing the square, we reorganized the equation into \( 25(y - 4)^2 - 10x - 519 = 0 \). Although it doesn't initially resemble the standard form of a parabola, observing the \( y^2 \) term without an accompanying \( x^2 \) term indicates the graph must be a parabola. Because the \( y \) terms are squared and the \( x \) terms are not, the parabola opens horizontally—left or right rather than up or down. In this case, since the coefficient of \( x \) is negative, the parabola opens to the left. Understanding these signs and coefficients is crucial in parabola classification as they determine the direction the parabola opens.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double napped cone. They include circles, ellipses, parabolas, and hyperbolas depending on the angle at which the plane cuts through the cone. Each of these conic sections can be represented by a general quadratic equation in two variables, which is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \).

The classification into circle, ellipse, parabola, or hyperbola depends on the values of the coefficients \( A \), \( B \), and \( C \), and their relationships. Specifically for a parabola, \( B^2 - 4AC = 0 \). In the given exercise, \( 25y^2 - 10x - 200y - 119 = 0 \) transforms into an equation with a squared term and no \( x^2 \) term, indicating that it represents a parabola. By using the techniques of completing the square and understanding the coefficients, we can classify the conic section as a parabola that opens to the left, which is a fundamental aspect of conic sections' study.