Problem 31 - Precalculus: Mathematics for Calculus

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$6 x^{2}+10 x y+3 y^{2}-6 y=36$$

Step-by-Step Solution

Answer

The conic is a hyperbola.

Step 1: Standard Form of Conic Equation

Rearrange the given conic section equation to clearly identify the coefficients of the quadratic terms:\[6x^2 + 10xy + 3y^2 - 6y - 36 = 0\]Recognize that this fits the general form: \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\] where \(A = 6\), \(B = 10\), \(C = 3\), \(D = 0\), \(E = -6\), \(F = -36\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) for a conic section is given by \[\Delta = B^2 - 4AC\]Substitute the values of \(A\), \(B\), and \(C\):\[\Delta = (10)^2 - 4(6)(3) = 100 - 72 = 28\]

Step 3: Determine the Type of Conic Section

Based on the value of the discriminant, identify the type of conic:- If \(\Delta = 0\), it is a parabola.- If \(\Delta > 0\), it is a hyperbola.- If \(\Delta < 0\), it is an ellipse.Since \(\Delta = 28 > 0\), the conic is a hyperbola.

Step 4: Graph the Conic Section

To confirm the result, graph the equation using graphing software or a calculator. Input the equation\[6x^2 + 10xy + 3y^2 - 6y = 36\]Observe the shape of the graph. You should see a pair of open curves bending away from each other, confirming that it is a hyperbola.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The different types of curves that can form are known as conic sections, and they include circles, ellipses, parabolas, and hyperbolas. These curves are fundamental in mathematics as they appear in various fields such as physics, engineering, and astronomy. Here are critical points about conic sections:

• Circle: Formed when the plane is perpendicular to the cone's axis, resulting in a perfectly round shape.
• Ellipse: An elongated circle, formed when the plane cuts through the cone at an angle, not perpendicular to the axis or through both nappes.
• Parabola: Occurs when the plane is parallel to the inclined plane of the cone, producing an open curve.
• Hyperbola: Arises when the plane cuts through both nappes of the cone, resulting in two open curves that extend infinitely.

Each conic section can be described with an equation in the general quadratic form:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\] By analyzing the coefficients and the discriminant (\(\Delta\)), we can determine the type of conic section it represents.

Hyperbola

A hyperbola is one of the distinct conic sections characterized by two separate and mirror-image curves known as branches. These branches never intersect each other. The hyperbola looks as if each branch is bending away from the center, which is different from circles or ellipses that loop back together:

• The standard form for a hyperbola, when it is not rotated, is given by the equation:\[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\]
• This equation implies there is an open form or `U` shape mirrored across different quadrants.
• Asymptotes: These are lines that the hyperbola approaches but never meets. They act as invisible boundaries for the hyperbola's branches.

The hyperbola is used in various scientific applications such as the design of reflective surfaces like satellite dishes and telescopes, enabling precise focusing capability. In our problem, using the discriminant calculated as \(\Delta = 28\) proves the conic is a hyperbola since \(\Delta > 0\). Understanding hyperbolas also allows us to interpret complex mathematical and real-world phenomena.

Graphing Conic Sections

Graphing conic sections visually reveals the properties defined by their equations. For graphing, it's crucial to rearrange the conic's equation to identify the specific type and shape. Here's how you can approach graphing:

• Start by ensuring the equation is in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
• Recognize the coefficients to calculate the discriminant. This tells you if the conic section is a circle, ellipse, parabola, or hyperbola.
• Visualize how the curve interacts with the coordinate axes:
• - For parabolas, observe a single open curve.
• - For ellipses and circles, look for closed, looped shapes.
• - For hyperbolas, identify two separate curve branches.

In the exercise provided, once you have identified the conic as a hyperbola using the discriminant, graph the equation on graphing software. This graphical representation will show two open, opposite-facing curves, confirming the hyperbola's distinct shape. By doing so, you not only verify the discriminant's result but also gain a better visual understanding of the conic section's nature.