Problem 31 - Algebra 2

Question

If two stones are thrown into a lake at different points, the points of intersection of the resulting ripples will follow a conic section. Suppose the conic section has the equation \(x^{2}-2 y^{2}-2 x-5=0\) Identify the shape of the curve.

Step-by-Step Solution

Answer

The curve is a hyperbola.

Step 1: Identify the Form of the Equation

The given conic section equation is \(x^2 - 2y^2 - 2x - 5 = 0\). This equation is a general form of a conic section \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Comparing coefficients, we get \(A = 1\), \(B = 0\), \(C = -2\), \(D = -2\), \(E = 0\), and \(F = -5\).

Step 2: Determine the Type of Conic Using Discriminant

The discriminant \(\Delta\) of a conic section is given by \(B^2 - 4AC\). Substituting our values, we have: \(0^2 - 4(1)(-2) = 8\). Since \(\Delta > 0\), the conic section is a hyperbola.

Key Concepts

Hyperbola Identification

Identifying a hyperbola involves analyzing the equation of a conic section. In our example, the given equation is \(x^2 - 2y^2 - 2x - 5 = 0\). To find out which type of conic it represents, you need to look at the coefficients of \(x^2\), \(y^2\), and any cross products like \(xy\). Here are some clues:

• If both the \(x^2\) and \(y^2\) terms are present with opposite signs, the conic is a hyperbola.
• If the signs are the same and non-zero, the conic is an ellipse or a circle.
• If only one of these terms is present, it could be a parabola.

In our equation, the terms \(x^2\) and \(y^2\) have opposite signs: \(+1\) for \(x^2\) and \(-2\) for \(y^2\). This tells us that the graph of the equation is a hyperbola.

Discriminant Method

The discriminant method is a mathematical approach that helps determine the type of conic section represented by a given equation. The key formula used is the discriminant \(\Delta = B^2 - 4AC\), derived from the general conic equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The value of \(\Delta\) influences the classification:

• \(\Delta = 0\): The conic is a parabola.
• \(\Delta > 0\): The conic is a hyperbola.
• \(\Delta < 0\): The conic is an ellipse or a circle.

By substituting the values from our conic equation into the formula, we obtained \(\Delta = 8\), since, \(0^2 - 4(1)(-2) = 8\). The positive \(\Delta\) confirms that our conic section is a hyperbola. Understanding the discriminant is a powerful tool for conic section identification.

General Form of Conic Sections

Conic sections can describe various curves and have a general equation format: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). This form is comprehensive and can reveal whether an equation is a circle, ellipse, hyperbola, or parabola. Each conic section has distinct characteristics:

• A circle occurs when \(A = C\) and \(B = 0\).
• An ellipse happens when \(A \, C > 0\) and \(B^2 - 4AC < 0\).
• A hyperbola takes shape when \(B^2 - 4AC > 0\).
• A parabola forms when \(B^2 - 4AC = 0\).

The coefficients \(A\), \(B\), \(C\), etc., dictate the properties and classification of the conic. By identifying these variables and utilizing methods like completing the square or using the discriminant, you can uncover the specific nature of any conic section captured by this general equation.