Problem 9 - Thomas Calculus

Question

Use the discriminant \(B^{2}-4 A C\) to decide whether the equations represent parabolas, ellipses, or hyperbolas. \(x y+y^{2}-3 x=5\)

Step-by-Step Solution

Answer

The equation represents a hyperbola.

Step 1: Identify the Type of Conic Section

The given equation is: \(xy + y^2 - 3x = 5\). First, we rearrange it to the standard form: \(xy + y^2 - 3x - 5 = 0\), which is of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, \(A = 0\), \(B = 1\), and \(C = 1\).

Step 2: Calculate the Discriminant

For a conic section given by \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is \(B^2 - 4AC\). Substitute \(A = 0\), \(B = 1\), \(C = 1\) into the formula: \[Discriminant = 1^2 - 4 imes 0 imes 1 = 1.\]

Step 3: Determine the Type of Conic Section Using the Discriminant

Compare the discriminant value to identify the conic section. If the discriminant \(B^2 - 4AC > 0\), the equation represents a hyperbola. For \(B^2 - 4AC = 0\), it represents a parabola. If \(B^2 - 4AC < 0\), it represents an ellipse. Since the discriminant is \(1 > 0\), it represents a hyperbola.

Key Concepts

Discriminant

The discriminant is an integral tool in identifying conic sections from a general quadratic equation. It's derived from the equation in its standard form: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The discriminant formula is given by \(B^2 - 4AC\). By calculating this value, we can determine the type of conic section:

• If \(B^2 - 4AC > 0\), the conic is a hyperbola.
• If \(B^2 - 4AC = 0\), it is a parabola.
• If \(B^2 - 4AC < 0\), it forms an ellipse.

The discriminant gives us crucial information regarding the nature and shape of the graph. In our example, the calculated value was 1, which is greater than 0, indicating a relation to a hyperbola.

Hyperbola

A hyperbola is one of the fascinating conic sections formed by the intersection of a plane with a double cone. It consists of two separate curves called branches. Unlike ellipses and circles, where points are close to a central point, hyperbolas stretch outwards. Key characteristics of hyperbolas include:

• Asymptotes: Straight lines that the branches will approach but never touch.
• Two connected components or branches, which create an open, unattached curve.
• The hyperbola can open either horizontally or vertically depending on the equation's orientation.

In our exercise, since the discriminant was positive, it confirms that the graph represents a hyperbola. Understanding hyperbolas and their properties helps us visualize how the graph behaves and interacts with its axes.

Equation Transformation

Transforming equations is often necessary to identify the type of conic section. For conic sections, the given equation needs rearrangement to fit the standard quadratic form: \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]The transformation involves moving terms, combining like terms, and sometimes completing the square, especially in more complex equations. This enables easier calculation of key components like the discriminant or completing the square to identify vertex form. During this process, remain attentive to the coefficients \(A\), \(B\), and \(C\) as their values are essential in computing the discriminant.In the given exercise, with the equation \(xy + y^2 - 3x = 5\), transformation showed \(A = 0\), \(B = 1\), and \(C = 1\) to present in the expected standard format. Proper transformation provides clarity and accuracy in further analysis and interpretation of the conic section. Keep practicing equation transformations to master identifying and graphing conic sections efficiently.