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Chapter 10: Problem 224

Determine which conic section is represented based on the given equation. \(-x^{2}+4 \sqrt{2} x y+2 y^{2}-2 y+1=0\)

Short Answer

Expert verified
The equation represents a hyperbola.

Step by step solution

01

Recognize the General Equation of Conic Sections

The given equation is \[-x^{2}+4\sqrt{2}xy+2y^{2}-2y+1=0\]. The general second-degree equation for a conic section is given by:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]. For our given equation:\[A = -1, \ B = 4\sqrt{2}, \ C = 2, \ D = 0, \ E = -2, \ F = 1.\]
02

Calculate the Discriminant

The discriminant \(\Delta\) of a conic section, which helps to identify the type of conic section, is calculated using:\[\Delta = B^2 - 4AC\].Substitute the values:\[B = 4\sqrt{2}, \ A = -1, \ C = 2\]\[\Delta = (4\sqrt{2})^2 - 4(-1)(2) = 32 + 8 = 40\].
03

Analyze the Discriminant

The nature of the conic section can be inferred from the discriminant:- If \(\Delta > 0\), the conic is a hyperbola.- If \(\Delta = 0\), the conic could be a parabola.- If \(\Delta < 0\), the conic could be an ellipse or circle.Since \(\Delta = 40 > 0\), the equation represents a hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hyperbola - Understanding the Shape
A hyperbola is a type of conic section that you can easily identify by its distinctive shape, consisting of two separate curves called "branches." This geometrical figure arises when a plane intersects both sheets of a double cone.

One key feature of a hyperbola is that it has two focal points. Unlike an ellipse where the sum of the distances from any point on the curve to its foci is constant, a hyperbola maintains the difference between these distances constant.

Hyperbolas have some properties which include, but are not limited to:
  • The transverse axis which is the line segment through the foci.
  • The conjugate axis which is perpendicular to the transverse axis at the center of the hyperbola.
  • Asymptotes, which are lines that the branches of the hyperbola approach but never U-E.
Discriminant - The Conic Identifier
In conics, the discriminant (\(\Delta\)) plays a crucial role in determining the type of conic section represented by a given quadratic equation. It's like a key that unlocks the understanding of whether the equation represents a circle, ellipse, parabola, or hyperbola.

The discriminant is given by the formula:
\[ \Delta = B^2 - 4AC \]Here, the values of \( A, B, \) and \( C \) are taken from the general equation:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

Let's explore how the discriminant helps:
  • If \(\Delta\) \(> 0\), the equation represents a hyperbola.
  • If \(\Delta\) \(= 0\), a parabola is formed.
  • If \(\Delta\) \(< 0\), you get an ellipse or a circle (if the coefficients also satisfy ellipse-specific conditions).
This clarity brought by the discriminant simplifies the process of identifying conic shapes significantly. It acts like a numerical litmus test!
General Equation of Conics - The Basis
The general equation of conic sections forms the foundation for understanding different types of conics. It's expressed as:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
This equation accommodates all conic sections - ellipses, hyperbolas, parabolas, and circles by changing the values of the coefficients \( A, B, C, \) and so on.

Breaking down the parts:
  • \( Ax^2 \) and \( Cy^2 \) dictate the direction and type based on their signs: both positive for ellipses, one negative for hyperbolas, etc.
  • \( Bxy \) introduces a rotation, affecting orientation.
  • \( Dx, Ex, \) and \( F \) shift the conic in the coordinate plane, elevating flexibility.
Understanding this equation allows one to comprehend the transformations and characteristics of conic sections thoroughly. It is the versatile language through which we can describe so many different curves in geometry, unlocking a wide range of analytical possibilities.

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Most popular questions from this chapter

Determine which of the conic sections is represented. $$4 x^{2}+14 x y+5 y^{2}+18 x-6 y+30=0$$

Write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. $$\frac{(x-2)^{2}}{100}+\frac{(y+3)^{2}}{36}=1$$

Identify the conic with focus at the origin, and then give the directrix and eccentricity. $$r=\frac{3}{2-\sin \theta}$$

Write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes. $$\frac{x^{2}}{49}-\frac{y^{2}}{81}=1$$

Graph the parabola, labeling vertex, focus, and directrix. $$(y-1)^{2}=\frac{1}{2}(x+3)$$
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