Discriminants for ellipse, parabola or hyperbola

In summary, a discriminant is a mathematical tool used to determine the type of conic section represented by an equation. It is calculated differently for ellipses, parabolas, and hyperbolas and can be used to graph these shapes. The value of the discriminant also tells us about the number and nature of solutions to the equation, allowing for real-world applications in various fields such as engineering, physics, and astronomy.
  • #1
duki
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Homework Statement



Use the discriminant to determine if the following are equations of an ellipse, parabola or hyperbola

[tex]6x^2-12xy+6y^2-5x+9=0[/tex]
[tex]5xy-4y^2+8x-3y+20=0[/tex]
[tex]x^2-9xy+5y^2-2=0[/tex]
[tex]10x^2-9xy+5y^2-2=0[/tex]
[tex]2y^2-10x+9y-8=0[/tex]

Homework Equations



The Attempt at a Solution



I got these answers:

[tex]0, parabola[/tex]
[tex]25, hyperbola[/tex]
[tex]44, hyperbola[/tex]
[tex]-119, ellipse[/tex]
[tex]0, parabola[/tex]

could someone check me on these? Thanks!
 
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  • #2


Shouldn't the third one be 81-4*1*5=61,hyperbola? Was that so hard you really needed a check?
 

FAQ: Discriminants for ellipse, parabola or hyperbola

1. What is a discriminant for an ellipse, parabola, or hyperbola?

A discriminant is a mathematical tool used to determine the type of conic section that is represented by a given equation. It is typically found in the quadratic formula and is used to determine the number and nature of the solutions (real, imaginary, or complex) to the equation.

2. How is the discriminant calculated for an ellipse, parabola, or hyperbola?

The discriminant for an ellipse, parabola, or hyperbola is calculated differently depending on the type of conic section. For an ellipse, it is calculated as b² - 4ac, for a parabola it is b² - 4ac, and for a hyperbola it is b² - 4ac. The values of a, b, and c can be found by rearranging the equation into standard form.

3. What does the discriminant tell us about the conic section?

The discriminant can tell us the number and nature of the solutions to the equation, which in turn can tell us the type of conic section. If the discriminant is positive, there are two distinct real solutions and the conic section is an ellipse. If the discriminant is zero, there is one real solution and the conic section is a parabola. If the discriminant is negative, there are two complex solutions and the conic section is a hyperbola.

4. Can the discriminant be used to graph conic sections?

Yes, the discriminant can be used to graph conic sections. By analyzing the value of the discriminant, we can determine the type of conic section and use this information to plot key points on the graph. For example, for an ellipse, we can use the center, vertices, and co-vertices to draw the shape. For a parabola, we can use the focus and directrix to plot key points. And for a hyperbola, we can use the center, vertices, and asymptotes.

5. Are there any real-world applications of discriminants for conic sections?

Yes, there are many real-world applications of discriminants for conic sections. For example, in engineering and architecture, discriminants can be used to design structures with specific shapes and dimensions. In physics, they can be used to analyze the motion of objects in conic paths. In astronomy, they can be used to study the orbits of planets and other celestial bodies. In all of these fields, the discriminant helps determine the type of conic section and provides valuable information for solving problems and making predictions.

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