Problem 27 - Precalculus: Mathematical for Calculus

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$2 x^{2}-4 x y+2 y^{2}-5 x-5=0$$

Step-by-Step Solution

Answer

The conic is a parabola, confirmed by graphing the equation.

Step 1: Write the General Equation

The given equation is \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\). This is of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, \(A = 2\), \(B = -4\), \(C = 2\), \(D = -5\), \(E = 0\), and \(F = -5\).

Step 2: Calculate the Discriminant

The discriminant for a conic section is given by \(B^2 - 4AC\). Substitute the values: \((-4)^2 - 4(2)(2) = 16 - 16 = 0\).

Step 3: Determine the Conic Type

Since the discriminant \(B^2 - 4AC = 0\), the given equation represents a parabola.

Step 4: Verify by Graphing

Use a graphing calculator or software to input the equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\). The graph should show a parabolic shape, confirming the type identified using the discriminant.

Key Concepts

Discriminant

The discriminant is a nifty tool that helps us figure out what type of conic section an equation represents. It's all about examining the coefficients in the equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The magic formula we use is \(B^2 - 4AC\). This value provides the clue:

• If \(B^2 - 4AC > 0\), the conic is a hyperbola.
• If \(B^2 - 4AC = 0\), the conic is a parabola.
• If \(B^2 - 4AC < 0\), the conic is an ellipse (or a circle if \(A = C\)).

In our scenario, we have the equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\). Plugging the coefficients into the discriminant formula, we compute \((-4)^2 - 4(2)(2) = 16 - 16 = 0\). Since the result is zero, we can be sure that the equation represents a parabola. Yay for discriminants making it crystal clear!

Graphing Conics

After using the discriminant to identify the conic, graphing it is a solid step to confirm your findings. There are various tools like graphing calculators or software available.
When graphing, input the equation carefully: \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\). Observe the form and layout. Different conics have unique appearances:

• Circle: A perfect round shape.
• Ellipse: Oval with two axes.
• Parabola: U or V shape depending on the orientation.
• Hyperbola: Two open curves mirroring each other.

For our equation, you should see a parabolic shape, validating our previous work with the discriminant. This visual representation is an educational and intuitive way to understand conics.

Parabola Identification

Identifying a parabola involves some understanding of its basic properties and distinguishing features. Unlike ellipses and hyperbolas, a parabola forms that well-known U or V shape. It consists of points that are equally distant from a point (called the focus) and a line (called the directrix).
In the general conic equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the parabola is special because when \(B^2 - 4AC = 0\), it indicates that there's no second degree term "competing" in orientation, thus keeping it as a single open curve.
Practically, if you transform or shift the parabola equation into its standard form \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\), you can see more clearly how it behaves in space. Parabolas are widely seen in physics (like projectile motion) and engineering; hence, mastering how to spot and sketch them is an essential skill.