Help with conic intersections using algebra

  • Thread starter Cacophony
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In summary, the two problems involve finding the intersection points of a line and a conic section without graphing, using only algebra. The first problem involves an ellipse with given points (-3,0) and (-2,2), while the second problem involves a hyperbola with given points (2,0) and (-3,-1). Further information is needed to solve these problems, as there are multiple possible solutions for each.
  • #1
Cacophony
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Homework Statement


Find the intersection points of: Line through (3,1) with slope -1/2, and ellipse through (-3,0) and (-2,2). No graphing just algebra.


The Attempt at a Solution


So far I have b=10/4 and a^2=9. I'm not sure how to finish this problem.

Homework Statement


Find the intersection point of: Line through (-1,-1) with slope 2, and hyperbola through (2,0) and (-3,-1). No graphs just algebra.



The Attempt at a Solution


So far I have b=1 and a^2=4. Again, not sure where to go from here. Any pointers?
 
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  • #2
Cacophony said:

Homework Statement


Find the intersection points of: Line through (3,1) with slope -1/2, and ellipse through (-3,0) and (-2,2). No graphing just algebra.


The Attempt at a Solution


So far I have b=10/4 and a^2=9. I'm not sure how to finish this problem.

Homework Statement


Find the intersection point of: Line through (-1,-1) with slope 2, and hyperbola through (2,0) and (-3,-1). No graphs just algebra.



The Attempt at a Solution


So far I have b=1 and a^2=4. Again, not sure where to go from here. Any pointers?

There are zillions of ellipses passing through (-3,0) and (-2,2). What haven't you told us? Same with the hyperbola.
 

FAQ: Help with conic intersections using algebra

1. What is the purpose of using algebra to find conic intersections?

The purpose of using algebra to find conic intersections is to solve for the points where two or more conic sections intersect. This allows us to accurately determine the coordinates of the intersection points and better understand the relationship between the conic sections.

2. What are the different types of conic sections?

The different types of conic sections are circles, ellipses, parabolas, and hyperbolas. These are all 2-dimensional shapes that can be formed by intersecting a plane with a cone at different angles.

3. How can I determine the number of intersection points between two conic sections?

The number of intersection points between two conic sections can be determined by analyzing the equations of the conic sections. For example, if the equations have the same degree and the same sign, there will be 0, 1, or 2 intersection points. If the equations have different degrees, there will be 2 or 4 intersection points.

4. Can algebra be used to find conic intersections in three-dimensional space?

Yes, algebra can be used to find conic intersections in three-dimensional space. However, the equations will involve three variables instead of two, making the calculations more complex. It may also be necessary to use additional geometric concepts, such as vectors, to accurately determine the intersection points.

5. Are there any shortcuts or tricks for solving conic intersections using algebra?

There are several shortcuts and tricks that can be used to solve conic intersections using algebra. These include using symmetry to simplify the equations, using substitution to eliminate variables, and using geometric properties of conic sections to determine the number of intersection points. It is important to have a strong understanding of algebraic concepts and conic sections to effectively use these shortcuts.

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