Problem 68 - Precalculus

Question

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 x^{2}-y^{2}+4 x+2 y-1=0$$

Step-by-Step Solution

Answer

The given equation represents a hyperbola.

Step 1: Identify the form of the equation

First, arrange the equation in its standard form, by grouping the \(x\) terms and \(y\) terms together.

Step 2: Complete the square

To further simplify and identify the form, you need to complete the square for \(x\) and \(y\). Thus the equation becomes \((2x+1)^2 - (y - 1)^2 = 0\).

Step 3: Classify the equation

Comparing this equation with the standard forms of conic sections, it is clear the equation represents a hyperbola because it is in the form \(a^2 - b^2 = 0\), where both \(a\) and \(b\) contains terms of \(x\) and \(y\).

Key Concepts

hyperbola

A hyperbola is one of the four types of conic sections, which are curves obtained by intersecting a plane with a double cone. Unlike the other conic sections, the hyperbola consists of two separate curves called branches. These branches mirror each other across both the x-axis and y-axis.
Understanding the properties of the hyperbola is crucial for solving equations involving them. Here are some key features to note about hyperbolas:

• A hyperbola has two focal points, and the difference of distances from any point on the hyperbola to the two foci is constant.
• The point at which the two branches appear closest is called a vertex. A hyperbola has two vertices, each pertaining to one branch.
• Asymptotes are crucial lines that guide the curvature of the hyperbola branches, ensuring they never actually connect.

When we encounter an equation and need to determine if it is a hyperbola, we look for terms involving both positive and negative squares, which cancel each other out—and identifying these in the form helps confirm the classification as a hyperbola.

completing the square

Completing the square is a powerful algebraic method that simplifies quadratic equations into a form that reveals more insights about the graph's nature, such as its vertex or center.

• This technique involves rearranging the quadratic equation so that the squared terms make a perfect square trinomial.
• For example, to complete the square for the term \(4x^2 + 4x\), you would factor as \(4(x^2 + x)\) and then adjust the equation to form a perfect square.

By completing the square, an equation like \(4x^2 - y^2 + 4x + 2y - 1 = 0\) can be transformed so that the equation clearly reveals its properties, such as the type of conic section it represents. Each perfect square trinomial corresponds to a squared binomial in the completed form, making it easier to classify the conic section correctly.

standard form of conic sections

The standard form of conic sections is essential for swiftly identifying the type of curve represented by a quadratic equation. Each type of conic section has its own standard form:

• A circle is represented by \( (x-h)^2 + (y-k)^2 = r^2 \) where \((h,k)\) is the center and \(r\) is the radius.
• An ellipse appears as \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), defining the eccentricity of the ellipse based on \(a\) and \(b\).
• A parabola is expressed as \( (y-k) = a(x-h)^2 \) for parabolas opening vertically or \( (x-h) = a(y-k)^2 \) for those opening horizontally.
• A hyperbola is recognized by \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \) to define its asymptotes and orientation.

The standard form helps to visualize and work with the geometrical properties of these curves by revealing their axes, centers, and distances directly in the equation. This clarity aids in solving the equations more effectively, allowing students to classify and understand the behavior of different conic sections.