Problem 28 - Precalculus with Calculus Previews

Question

In Problems \(23-28,\) use the discriminant to identify the conic without actually graphing. $$ 3 x^{2}+2 \sqrt{3} x y+y^{2}-2 x+2 \sqrt{3} y-4=0 $$

Step-by-Step Solution

Answer

The conic section is a parabola.

Step 1: Identify the Coefficients

The given equation is in the standard form of a conic section, which is written as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Identify the coefficients: \[ A = 3, \ B = 2\sqrt{3}, \ C = 1, \ D = -2, \ E = 2\sqrt{3}, \ F = -4. \]

Step 2: Write the Formula for the Discriminant

The discriminant for a conic section is given by the formula \(\Delta = B^2 - 4AC\). This value helps determine the type of conic section represented by the equation.

Step 3: Calculate the Discriminant

Substitute the coefficients into the discriminant formula: \[\Delta = (2\sqrt{3})^2 - 4 \cdot 3 \cdot 1 = 4 \cdot 3 - 12 = 12 - 12 = 0.\] Thus, the discriminant \(\Delta = 0\).

Step 4: Interpret the Discriminant

When the discriminant \(\Delta = 0\), the conic section is a **parabola**. The specific relationship between \(A\), \(B\), and \(C\) confirms it, as parabolas can be identified when \(\Delta = B^2 - 4AC = 0\).

Key Concepts

Discriminant

The discriminant is a crucial concept in understanding conic sections. It is a value derived from the coefficients of a conic section equation. Using the formula \(\Delta = B^2 - 4AC\), we can determine the nature of the conic section without needing to graph it. Here’s how it works:

• If \(\Delta > 0\), the conic section is a hyperbola.
• If \(\Delta = 0\), it is a parabola.
• If \(\Delta < 0\), it is an ellipse (or a circle if \(A = C\)).

Knowing the discriminant gives you a simple way to identify whether you're dealing with a parabola, ellipse, or hyperbola just by looking at the equation's coefficients. This is particularly useful when graphing is not feasible or when you need a quick assessment of the conic type.

Equation of Conic Sections

Conic sections are defined by the general quadratic equation \[\[\begin{equation}Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\end{equation}\]\]This equation can represent different geometrical shapes: parabolas, ellipses, and hyperbolas. To identify the shape, focus on the coefficients \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\). Each type of conic section has specific characteristics based on these coefficients:

• The term with \(Bxy\) tells us about the rotation of the conic section. If \(B e 0\), the conic is usually rotated from the standard axis.
• The ratio of \(A\) and \(C\) helps in identifying the conic. For instance, if \(A = C\) and \(B = 0\), it's a circle.

Understanding the equation's role is key to delving deeper into geometry and trigonometry, as these shapes are foundational in many math areas. The equation's simplicity allows it to act as a guide in quickly identifying the conic, mimicking how we use fingerprints to identify individuals.

Parabola

A parabola is one of the simplest forms of a conic section and is defined by having the discriminant \(\Delta = 0\) when using the coefficients \(A\), \(B\), and \(C\). Here’s what makes a parabola unique:

• It can be represented as either \(y=ax^2+bx+c\) (a vertical parabola) or \(x=ay^2+by+c\) (a horizontal parabola).
• Parabolas have a distinctive U-shape, which opens upwards, downwards, left, or right depending on the equation.
• They have an axis of symmetry, which is a line passing through the vertex turning it into a mirror image on either side.
• The vertex is the highest or lowest point depending on the direction of the opening.
• They occur naturally in physics, particularly in projectile motion, since the path of projectiles forms a parabolic trajectory.

Grasping the concept of a parabola not only helps in solving mathematical problems but also in understanding various real-world phenomena. The symmetry and predictable path of a parabola are what make it so useful and widely applicable.