Easy conic section question

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In summary, the equation x^2-y^2-9=0 is a hyperbola centered at the origin with lines of symmetry at x=0 and y=0. To find the domain, you can either convert the equation into standard form and use the vertices as a guide, or solve for x and y separately and look for any undefined regions. The range can be found in a similar manner.
  • #1
zaboda42
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Please Help! Easy conic section question!

1. Graph each equation. Identify the conic section and its lines of symmetry. Then find the domain and range.



2. x^2 - y^2 - 9 = 0



3. I have no idea how to do this. I know it's a hyperbola because i solved for "y" and graphed it in my calculator, but i have no idea how to find the domain/range and values. Help!
 
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  • #2


That is a hyperbola centered at the origin. Add +9 to both sides of the equation and then divide both sides by +9. I do not remember how to handle the rest of the graphing but the methods and characteristics should be well discussed and exemplefied in your book.
 
  • #3


There are two ways I can think of to answer this problem. The first is much easier, but only useful if you can visualize the hyperbola's rough shape.

1) Convert it into: [tex]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/tex]
With this hyperbola, if you understand its shape, you can find its domain such that [tex]x\leq a[/tex] and [tex]x\geq a[/tex] since the intersections are at [tex]x=\pm a[/tex]


2) Solve for y and find (if any) undefined regions, such as divisions by 0 or negatives in the square roots. This will give you the domain. (no undefined regions means that the domain is all x)
For the range, solve for x and do the same procedure.
 
  • #4


You shouldn't have needed your calculator. [itex]x^2- y^2= 9[/itex] or [itex]x^2/9- y^2/9= 1[/itex] is the standard form for a hyperbola with center at (0,0) and vertices at (3,0) and (-3,0). Knowing those vertices should give you the domain immediately.
 

FAQ: Easy conic section question

1. What are conic sections?

Conic sections are geometric curves that are formed by the intersection of a cone and a plane. The cone can be visualized as a three-dimensional object with a circular base that tapers to a point. When a plane intersects the cone at different angles and distances from the base, different types of conic sections are formed, including circles, ellipses, parabolas, and hyperbolas.

2. What are some real-life applications of conic sections?

Conic sections have many practical uses in fields such as engineering, physics, and astronomy. For example, parabolic mirrors are used in telescopes and satellite dishes to focus and reflect light. The elliptical shape of orbits around the sun can be described using conic sections. Conic sections also have applications in designing bridges, roller coasters, and other structures.

3. How do you identify the type of conic section from its equation?

The general equation of a conic section is in the form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. By examining the coefficients of the x^2 and y^2 terms, you can determine the type of conic section. If A and C have different signs (positive and negative), the conic section is an ellipse. If A and C are both positive or both negative, it is a hyperbola. If one of the coefficients is 0 and the other is non-zero, it is a parabola. If A and C are both 0, it is a circle.

4. What is the focus-directrix property of conic sections?

The focus-directrix property states that for a given conic section, there is a point (focus) and a line (directrix) that have a special relationship with all points on the curve. For ellipses and hyperbolas, the distance from any point on the curve to the focus is equal to the distance from that point to the directrix. For parabolas, the focus is located at the same distance from the vertex as the directrix.

5. How are conic sections related to the eccentricity of a curve?

The eccentricity of a conic section is a measure of how "squished" or elongated the curve is compared to a circle. It is calculated by dividing the distance from the center to a focus by the distance from the center to a point on the curve. The closer the eccentricity is to 0, the more circular the curve is, and the closer it is to 1, the more elongated it is. Circles have an eccentricity of 0, ellipses have an eccentricity between 0 and 1, parabolas have an eccentricity of 1, and hyperbolas have an eccentricity greater than 1.

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