CHECK Conics question on general form of conic equation

[aisha]

$$6x^2 + 2y^2 - 9x +14y -68=0$$

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?

 

[Data]

What are you calling $a$ and what are you calling $b$?

 

[aisha]

Well when in general form a conic's equation is in the form

$$ax^2 + by^2 + 2gx + 2fy + c = 0$$ general form

so I am using the first two terms in the given equation as a and b

 

[Data]

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 $a \leftrightarrow b$ makes the semi-major axis horizontal.

 

[aisha]

horizontal ellipse means that the major axis is horizontal and

vertical ellipse means that the major axis is vertical.

I don't understand or know the semi-terms yet is my answer correct?

 

[Data]

in that case it's vertical to start and horizontal when you switch them.

 

[aek]

yeah that's about right Data

 

[aisha]

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?

 

[Data]

Assume $ab>0$. Then

$$a>b$$

implies that it's vertical.

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

$$a(x-\gamma)^2 + b(y-\alpha)^2 = \lambda$$

so dividing out $ab$ gives

$$\frac{(x-\gamma)^2}{b} + \frac{(y-\alpha)^2}{a} = \frac{\lambda}{ab}$$

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