Problem 41 - Precalculus

Question

(a) use the discriminant to classify the graph of the equation, (b) use the Quadratic Formula to solve for \(y\) and (c) use a graphing utility to graph the equation. $$x^{2}-6 x y-5 y^{2}+4 x-22=0$$

Step-by-Step Solution

Answer

The discriminant of the equation is 56, so it represents a hyperbola. Solve for \(y\) gives us two solutions in terms of \(x\), which when plotted form a hyperbola on the graphing utility.

Step 1: Calculate the Discriminant

The given equation is of the form \(Ax^2 + Bxy + Cy^2 + Dx - E = 0\). Here \(A = 1, B = -6, C = -5\). To calculate discriminant, use the formula \(B^2 - 4AC\). Substituting the values, the discriminant is \((-6)^2 - 4*1*-5 = 36 + 20 = 56.

Step 2: Classify the Graph

The sign of the discriminant can be used to classify conic sections. If \(B^2 - 4AC > 0\), the graph represents a hyperbola, if \(B^2 - 4AC = 0\) it represents a parabola, and if \(B^2 - 4AC < 0\) it represents an ellipse or circle. Since our discriminant is 56, the graph of the equation will represent a hyperbola.

Step 3: Solve for \(y\) using Quadratic Formula

To solve for \(y\), rearrange the equation into following quadratic equation \(5y^2 + 6xy - x^2 - 4x + 22 = 0\). Let \(y = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\) where \(A=5, B=6x, C= -x^2-4x+22\). By substituting these values into the formula, we get \(y = (-6x \pm \sqrt{(6x)^2 - 4*5*(-x^2 - 4x + 22)}) / (2*5)\) for different values of \(x\).

Step 4: Graph the equation

To graph the equation, input the obtained values of \(y\) depending on \(x\) in a graphing utility. Mark the points on \(y\) corresponding to each \(x\) value and then join these points to create a hyperbola graph.

Key Concepts

Hyperbola

A hyperbola is a fascinating shape that you might come across while studying conic sections. It's important to understand how it relates to the equation we're working with. A hyperbola consists of two separate curves, which resemble open-ended ellipses.
They are further apart as they extend outward, creating a mirror-like appearance on a graph.A crucial way to identify a hyperbola in a quadratic equation is through the discriminant. The discriminant is calculated using the formula \(B^2 - 4AC\), and it helps classify different conic sections without graphing the equation. For our equation \(x^{2}-6xy-5y^{2}+4x-22=0\), the discriminant is \(56\), which is positive. That indicates that the graph of this equation is a hyperbola. So, a positive discriminant \(>0\) means we are dealing with a hyperbola when analyzing conic sections.
Remember these key points to recognize a hyperbola:

• Two separate, symmetric curves.
• The discriminant is positive \(>0\).
• Looks like two "bowls" facing away from each other.

Recognizing these patterns will help you easily identify a hyperbola in equations.

Quadratic Formula

The quadratic formula stands tall in the toolkit of algebraic operations. It allows us to find the values of a variable within a quadratic equation efficiently. This formula is especially useful when solving equations of the form \(Ax^2 + Bx + C = 0\).The general formula is:\[y = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\]For our specific equation—after rearranging—we solve for \(y\) using:\[y = \frac{-6x \pm \sqrt{(6x)^2 - 4*5*(-x^2 - 4x + 22)}}{2*5}\]This allows us to find the values of \(y\) in terms of \(x\). It's essential to understand that each part of the formula plays a vital role:

• The term \(-B\) helps adjust the equation by the slope factor.
• The square root handles both possible trajectories (and solutions) \(\pm \sqrt{B^2 - 4AC}\).
• Finally, the division by \(2A\) normalizes the result according to \(A\).

By plugging specific values of \(x\) into this formula, we'll derive precise values for \(y\). This method makes it possible to tackle complex equations by delivering exact solutions regardless of the complexities involved.

Conic Sections

Conic sections are a group of shapes created by intersecting a plane with a cone. They are fundamental in the study of geometry and algebra. The main conic sections include circles, ellipses, parabolas, and hyperbolas.The type of conic section we observe largely depends on the discriminant \(B^2 - 4AC\):

• Ellipse: When the discriminant is negative \(<0\).
• Parabola: When the discriminant is zero \(=0\).
• Hyperbola: When the discriminant is positive \(>0\).

So, depending on the values of \(A\), \(B\), and \(C\), we can determine the shape of the conic section by evaluating the discriminant.Conic sections are not just theoretical; they have real-world applications. These shapes can be seen in planetary orbits, architectural designs, and even in light physics due to the reflective properties. For example, the hyperbolas' mirrors are used in radio telescopes due to their unique reflective traits.Recognizing and understanding conic sections helps in visualizing and solving complex algebraic equations, as they can be a useful tool across various fields.