1. ## hyperbola or parabola?

Find the conic section represented by the equation: x=y^2+10

I think this is a parabola. But how do you tell if the equation is a hyperbola or not?

2. Originally Posted by RenSully
Find the conic section represented by the equation: x=y^2+10

I think this is a parabola. But how do you tell if the equation is a hyperbola or not?
Put it in the standard form $\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

Compute $\displaystyle B^2 - 4AC$: if the number is negative, it's an ellipse; if it's zero, it's a parabola; if it's positive, it's a hyperbola.

Here $\displaystyle B^2 - 4AC = 0^2 - 4(0)(1) = 0$

3. That helps a lot, thank you.

How do you know if it's a circle? My guess is that the coefficients on x^2 and y^2 are the same. Is that right?

4. That helps a lot, thank you.

How do you know if it's a circle? My guess is that the coefficients on x^2 and y^2 are the same. Is that right?

5. opps, didn't mean to post twice.

Here is an example question about the circles:

Find an equation that represents a circle.

6. Originally Posted by RenSully
Find an equation that represents a circle.

Above I mentioned $\displaystyle B^2 - 4AC$. If there's no B, and A = C, then it's a circle. In other words a circle is just a special kind of ellipse.

7. Without the "xy" term it is particularly easy to tell:

If there is either $\displaystyle x^2$ or $\displaystyle y^2$ but not both, the graph is a parabola.

If the coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ (when they are on the same side of the equation) are of opposites sign, it is hyperbola.

If the coefficients of $\displaystyle x^2$ and $\displaystyle y^2$ (when they are on the same side of the equation) are of the same sign, it is an ellipse.
Special case: if the coefficients are exactly the same, then the graph is a circle.

8. Originally Posted by RenSully
Find the conic section represented by the equation: x=y^2+10

I think this is a parabola. But how do you tell if the equation is a hyperbola or not?
You can tell iff an equation is a hyperbola, ellipse, or parabola by making the right side of the equation 1, analyzing the coefficients of the variable terms, and then observing whether or not the sign seperating the terms is positive or negative.

In this case...move y^2 to the other side

$\displaystyle x-y^2=10$

Divide both sides by ten to analyze the coefficients

$\displaystyle \frac{1}{10}x-\frac{1}{10}y^2=1$

Since the coefficients are equal and only one of the variable terms is quadratic, this is a parabola.

9. what if there is an Bxy in the equation?
like:
Classify the conic represented by the equation

I would think this is a hyperbola, but with the -4xy I am not sure.

10. Originally Posted by RenSully
Classify the conic represented by the equation

$\displaystyle x^2 - 4xy - 20y^2 + 2x + 6y - 6 = 0$

I would think this is a hyperbola, but with the -4xy I am not sure.
A = 1
B = -4
C = -20

$\displaystyle B^2 - 4AC = (-4)^2 - 4(1)(-20) = 16 + 80 = 96$

Probably this is in an upcoming chapter for you, but the xy term can be "rotated out" of the equation using a substitution. Doesn't matter, though. The $\displaystyle B^2 - 4AC$ catches them every time. I guess you could say it helps you 'discriminate' between the types ... which is why it's called the 'discriminant.'