CHECK Conics question on general form of conic equation

In summary, the given equation represents an ellipse because a does not equal b and ab>0. The value of "a" to transform the conic into a circle would be 2, while the value of "b" would be 6. If the values of "a" and "b" were interchanged, the ellipse would change from being vertical to horizontal.
  • #1
aisha
584
0
[tex] 6x^2 + 2y^2 - 9x +14y -68=0 [/tex]

a) which conic is represented by the equation why?

I think the ellipse is represented by the equation because a does not = b and ab >0

b)What value of "a" would transform the conic into a circle?

I think when a=b and ab>0 then the conic will be transformed into a circle therefore the value of "a" will be 2 in order for the conic to transform into a circle.

c)What value of "b" would transform the original conic into a circle?

I think when b=a and ab>0 then the conic will be transformed into a circle. Therefore the value of "b" to transform the original conic into a circle would be 6.

d) What change would take place if the value of "a" and "b" were interchanged?

I think if the values of "a" and "b" were interchanged then the ellipse would change from being a horizontal ellipse to a vertical ellipse.

Can some1 please help me out and tell me if any of my answers are wrong? :redface:
 
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  • #2
What are you calling [itex]a[/itex] and what are you calling [itex]b[/itex]?
 
  • #3
Well when in general form a conic's equation is in the form

[tex] ax^2 + by^2 + 2gx + 2fy + c = 0 [/tex] general form

so I am using the first two terms in the given equation as a and b
 
  • #4
All of your answers are okay, except for the last one. I don't know what you mean by a horizontal or vertical ellipse. As given, the semi-major axis of the ellipse is vertical, so [itex] a \leftrightarrow b[/itex] makes the semi-major axis horizontal.
 
  • #5
horizontal ellipse means that the major axis is horizontal and

vertical ellipse means that the major axis is verticle.

I don't understand or know the semi-terms yet is my answer correct?
 
  • #6
in that case it's vertical to start and horizontal when you switch them.
 
  • #7
yeah that's about right Data
 
  • #8
In one of my other posts someone wrote

If the denominator attached to the x is less than that of the y then it will be vertical, (as there is less of a distance between the x intercepts that the y).

The semi-major axis will be the root of the largests denominator
The semi-minor axis will be the root of the smallest denominator
The coordinates of the center will be (h,k) for and elispse in the form (just a simple translation)

but this was if the equation was in standard form.

Can you explain to me how you know if the ellipse is vertical or horizontal looking at the general form of the equation?
 
  • #9
Assume [itex]ab>0[/itex]. Then

[tex]a>b[/tex]

implies that it's vertical.

Remember when you complete the squares you'll get something in the form

[tex] a(x-\gamma)^2 + b(y-\alpha)^2 = \lambda[/tex]

so dividing out [itex]ab[/itex] gives

[tex] \frac{(x-\gamma)^2}{b} + \frac{(y-\alpha)^2}{a} = \frac{\lambda}{ab}[/tex]

and thus using the rule you posted above, the ellipse is vertical if [itex]a>b[/itex] and horizontal if [itex]b>a[/itex] (and a circle if [itex]a=b[/itex]).
 

1. What is the general form of a conic equation?

The general form of a conic equation is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, C, D, E, and F are constants and x and y are variables.

2. How do you classify conic equations?

Conic equations can be classified based on the value of the discriminant B2 - 4AC. If the discriminant is positive, the conic is an ellipse. If it is negative, the conic is a hyperbola. If it is zero, the conic is a parabola.

3. How do you graph a conic equation?

To graph a conic equation, you can plot points by substituting different values for x and y into the equation and then connecting them with a smooth curve. Alternatively, you can use the properties of the conic to sketch the graph, such as the distance between points on an ellipse or the direction of the branches on a hyperbola.

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

The focus-directrix property states that for any point on the conic, the distance to the focus is equal to the distance to the directrix. This property is used to define the shape of the conic and can be used to find the equation of the conic given the focus and directrix.

5. How are conic equations used in real life?

Conic equations have many real-life applications, such as in astronomy to describe the orbits of planets and satellites, in engineering to design curved structures and lenses, and in physics to model motion and trajectories of objects. They are also used in everyday objects like satellite dishes, parabolic mirrors, and curved bridges.

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