Problem 12 - College Algebra

Question

For the following exercises, determine which conic section is represented based on the given equation. $$4 x^{2}+9 x y+4 y^{2}-36 y-125=0$$

Step-by-Step Solution

Answer

The conic section is a hyperbola.

Step 1: Identify the type of conic

The given equation is in the general form of a conic: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] Substitute the coefficients from the given equation: \( A = 4, B = 9, C = 4, D = 0, E = -36, F = -125 \).

Step 2: Use the discriminant to determine the conic section

The discriminant \( \Delta \) for a conic section is given by the formula: \[ \Delta = B^2 - 4AC \]Substitute \( B = 9, A = 4, C = 4 \):\[ \Delta = 9^2 - 4(4)(4) = 81 - 64 = 17 \]Since \( \Delta > 0 \), the conic section is a hyperbola.

Key Concepts

Discriminant of Conic Sections

The discriminant is a valuable tool for determining the type of conic section represented by a quadratic equation in two variables. The general form of such an equation is \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]The discriminant formula is given by: \[ \Delta = B^2 - 4AC \]This mathematical expression helps us understand the relationship between the coefficients of the equation. Here is what the discriminant can reveal:

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

Using the discriminant simplifies the process of identifying the conic sections without needing to rewrite or graph the equations. For students, understanding these relationships among the coefficients is crucial for analyzing and working with conic equations efficiently.

Identifying Conic Sections

Conic sections are the curves obtained by the intersection of a plane and a double-napped cone. Four major types of conic sections exist: circle, ellipse, parabola, and hyperbola. Identifying which conic section an equation represents is vital:Equations can be recognized based on their coefficients and terms:

• Circle: The equation has the form \( x^2 + y^2 \), with equal coefficients for \( x^2 \) and \( y^2 \) and no \( xy \) term.
• Ellipse: Similar to a circle, but the coefficients of \( x^2 \) and \( y^2 \) are not equal.
• Parabola: The equation includes either the \( x \) or \( y \) term alone, not both squared.
• Hyperbola: The equation has both \( x^2 \) and \( y^2 \) terms, and often includes an \( xy \) term, with the discriminant indicating a hyperbola when \( \Delta > 0 \).

Understanding how to identify these conic sections based on their standard forms and characteristics is key for solving many algebraic and geometric problems.

Hyperbola

A hyperbola is one of the four primary types of conic sections, appearing when a plane intersects both nappes of a double cone. It might look complex, but understanding its features helps it become more approachable.Here are some characteristics of hyperbolas:

• They consist of two disconnected curves called "branches".
• The branches open either along a horizontal axis or a vertical axis.
• To standardize its form, the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) describes a hyperbola centered at the origin with horizontal branches.
• The focal points within each branch always lie outside the hyperbola, unlike an ellipse.

Recognizing a hyperbola from its equation allows you to predict and draw its unique structure. By understanding the discriminant and identifying the conic sections, hyperbolas become more than just abstract concepts; they translate into tangible mathematical structures with fascinating properties.