# Thread: How can I determine this is a parabola? (by finding its standard form)

1. ## How can I determine this is a parabola? (by finding its standard form)

The equation is:

3x^2-20y^2=80y+48

I know it is a hyperbola once I check with Wolfram Alpha. But, I need a definite way of knowing it is.

I tried setting it up in the standard form by trying to isolate 48.

3x^2-20y^2-80y=48

Then divide by 48 to make the right side equal 1 like the standard form for it is.

3x^2/48-(20y^2-80y)/48=1

But that is still not its standard form because I don't know what y^2 is... how can I get it to be just y^2?

2. Originally Posted by thyrgle
The equation is:

3x^2-20y^2=80y+48

I know it is a hyperbola once I check with Wolfram Alpha. But, I need a definite way of knowing it is.

I tried setting it up in the standard form by trying to isolate 48.

3x^2-20y^2-80y=48

Then divide by 48 to make the right side equal 1 like the standard form for it is.

3x^2/48-(20y^2-80y)/48=1

But that is still not its standard form because I don't know what y^2 is... how can I get it to be just y^2?

You first need to complete the square in y: $\displaystyle 20y^2+80y=20(y^2+4y)=20(y+2)^2-80$ , and then:

$\displaystyle \displaystyle{3x^2-20y^2=80y+48\Longrightarrow 3x^2-20(y+2)^2+80=48\Longrightarrow 3x^2-20(y+2)^2=-32\Longrightarrow$

$\displaystyle \frac{(y+2)^2}{8/5}-\frac{x^2}{32/3}=1$

Tonio

3. Originally Posted by thyrgle
3x^2-20y^2=80y+48
Re-arranging:
20y^2 + 80y - 3x^2 + 48 = 0
Clearly of form Ax^2 + Bx + C = 0

4. "How can I determine this is a parabola?"

first note that tonio shows how to derive 'standard' form. Wilmer had shown'general form'.

full conic general form is $\displaystyle Ax^2+Bxy+Cy^2+Dx+Ey+F$

a quick way to determine if you have parabola,hyperbola or ellipse is to look at A and C terms.

if there is only one second power term you have a parabola.
if C is negative you have a hyperbola.
if A and C are positive you have ellipse.

in standard form just look at the sign in between the terms '-'=hyperbola '+'=ellipse

to determine algebraically with NO B term.
A=C circle
AC > 0 = ellipse
AC = 0 = parabola
AC < 0 = hyperbola

to determine algebraically with B term use the discriminate.
$\displaystyle B^2-4AC$
discriminate < 0 = ellipse
discriminate = 0 = parabola
discriminate > 0 = hyperbola

5. Originally Posted by Wilmer
Re-arranging:
20y^2 + 80y - 3x^2 + 48 = 0
Clearly of form Ax^2 + Bx + C = 0
What????

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7. Hello, thyrgle

Is this a parabola? . $\displaystyle 3x^2-20y^2\:=\:80y+48$
Answer: no.

Skoker explained the equation of the general conic equation,
. . which includes conics which have been "rotated".

If there is no xy-term, the conic is "horizontal" or "vertical".
. . Its axis is parallel to a coordinate axis.

Then the equation can be "eyeballed" to identify the conic.

The general equation of this conic is: .$\displaystyle Ax^2 + By^2 + Cx + Dy + E \:=\:0$

We are concerned with $\displaystyle A$ and $\displaystyle B$ only, the coefficients of $\displaystyle x^2$ and $\displaystyle y^2.$

If $\displaystyle A = 0$ or $\displaystyle B = 0$ (but not both), we have a parabola

If $\displaystyle A = B$, we have a circle.

If $\displaystyle A \ne B$ and $\displaystyle A,B$ have the same sign: ellipse.

If $\displaystyle A,B$ have opposite signs: hyperbola.

Note: the above forms include degenerate and imaginary conics.

For example: .$\displaystyle x^2 + y^2 \:=\:0$ is a degenerate circle.

And: .$\displaystyle 4x^2 + y^2 + 9\:=\:0$ is an imaginary ellipse.