Problem 58 - College Algebra with Corequisite Support

Question

Given \(3 x^{2}+k x y+4 y^{2}-6 x+20 y+128=0\) find \(k\) for the graph to be a hyperbola.

Step-by-Step Solution

Answer

k > 4\sqrt{3} or k < -4\sqrt{3}.

Step 1: Identify the General Conic Equation

The given equation is: \[3 x^{2} + k x y + 4 y^{2} - 6 x + 20 y + 128 = 0.\] This equation is of the form of a general conic section: \[A x^{2} + B x y + C y^{2} + D x + E y + F = 0,\] where \( A = 3, B = k, C = 4, D = -6, E = 20, F = 128 \). In this conic form, the type of conic section can be determined by calculating the discriminant \( ext{Δ} = B^{2} - 4AC \).

Step 2: Evaluate Hyperbola Condition

For a conic section to be a hyperbola, the discriminant must satisfy the condition \[B^{2} - 4AC > 0.\] Let's substitute the known values: \[B = k, A = 3, C = 4.\] The discriminant becomes \[k^{2} - 4(3)(4) > 0.\]

Step 3: Simplify the Discriminant

Substitute the values for \(A\) and \(C\) into the inequality: \[k^{2} - 4(3)(4) > 0\] Simplify: \[k^{2} - 48 > 0.\]

Step 4: Solve the Inequality

To solve the inequality \(k^{2} - 48 > 0\), add 48 to both sides to obtain: \[k^{2} > 48.\] Take the square root of both sides: \[|k| > \sqrt{48}.\] Simplify \(\sqrt{48}\) as \(4\sqrt{3}\). Thus, \(|k| > 4\sqrt{3}.\)

Step 5: Determine the Solution for k

Since the inequality \(|k| > 4\sqrt{3}\) is established, we solve for \(k\):The solution implies: 1. \(k > 4\sqrt{3}\) 2. \(k < -4\sqrt{3}\). Thus, \(k\) must be either greater than \(4\sqrt{3}\) or less than \(-4\sqrt{3}\) for the equation to represent a hyperbola.

Key Concepts

Hyperbolas in Conic Sections

In geometry, a hyperbola is one type of conic section that can appear in equations. Conic sections are curves obtained by intersecting a cone with a plane at various angles.
The main types are circles, ellipses, parabolas, and hyperbolas. A hyperbola is composed of two "branches" that mirror each other, and it curves away from a central point or foci.To recognize a hyperbola analytically, check the equation of the conic form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, \]by using the discriminant \( \Delta = B^2 - 4AC \).

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

Understanding these conditions is crucial for determining what type of conic section you have.

Understanding the Discriminant in Conics

The discriminant is a critical part in identifying the nature of a conic section. It is a specific expression that utilizes coefficients from the equation of a conic section.For an equation in the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, \]the discriminant \( \Delta \) is calculated as \[ \Delta = B^2 - 4AC. \]This formula helps us decide the shape that the conic section will take.

• If \( \Delta > 0 \), the presence of two distinct intersecting paths indicates a hyperbola.
• If \( \Delta = 0 \), it represents a parabola, often a single path.
• If \( \Delta < 0 \), concentric and bounded paths reveal an ellipse or potentially a circle.

The discriminant's value directly informs us about the symmetry and type of the curve resulting from the equation.

Solving Inequalities Involving Conic Discriminants

Inequalities appear frequently when determining conditions for curves like hyperbolas. Solving these requires a bit of algebraic manipulation.Starting with\[ k^2 - 48 > 0, \]our goal is to find intervals of \( k \) that satisfy this inequality.

• First, we rearrange to isolate the squared term on one side: \( k^2 > 48 \).
• Next, take the square root of both sides. This results in two possible inequalities:
  \( k > \sqrt{48} \) or \( k < -\sqrt{48} \).
• Simplify the square root: \( \sqrt{48} = 4\sqrt{3} \).

Thus, the conditions become:
\( k > 4\sqrt{3} \) or \( k < -4\sqrt{3} \).
These inequalities indicate the range within which \( k \) can vary to maintain the hyperbolic nature of the graph. Solving these inequalities precisely allows students to conclude the solution and verify the specific requirements needed for each type of conic section.