Conic Sections Question

**sleigh** · March 24th 2010, 02:34 PM

Hi. Is there any way to know if a ellipse in the form $ax^2+by^2+cx+dy+e=0$ has a horizontal or vertical major axis without changing into standard form. For example, if I have the ellipse $2x^2+3y^2-x+6y-2=0$ can I tell if it is horizontal or vertical without changing it into standard form. I would think that the quadratic term ( $ax^2, by^2$ ) with the lower coefficent would be over $a^2$ , allowing one to determine whether it was horizontal or vertical, but I could be wrong. Could someone tell me if I am correct? Thanks!

**hollywood** · March 25th 2010, 12:11 AM

You're on the right track. The coefficients of $x^2\text{ and }y^2$ (a and b) are all that matter. If a>b, it's one, and if a<b it's the other. Of course, if a=b, it's a circle.

I'll let you figure out which one is which. Look at the equation $ax^2+by^2=\text{constant}$ when x=0 and when y=0.

Of course, a and b have to be positive, and it's possible that you'll end up with $ax^2+by^2=-\text{constant}$ after completing the square in x and y, so your equation would have no solutions.

Post again in this thread if you're still having trouble.

**sleigh** · March 25th 2010, 12:25 PM

I think that whichever one, $x^2$ or $y^2$ has a smaller coefficent would be over $a^2$ . In this ellipse: $2x^2+4y^2=9$ , dividing by 9 yields $\frac{2x^2}{9} + \frac{4y^2}{9}=1$ . Moving the coefficents of $x^2$ and $y^2$ into the denominator, I get $\frac{x^2}{\frac{9}{2}}+\frac{y^2}{\frac{9}{4}}=1$ Am I correct in assuming that if the coefficent in front of $x^2$ is smaller than the coefficent in front of $y^2$ , then the ellipse is horizontal, and if the coefficent in front of $y^2$ is larger than the coefficent in front of $x^2$ , it is vertical? Thank you for your help!

**hollywood** · March 25th 2010, 09:28 PM

You can work straight from $2x^2+4y^2=9$ . If x=0, y is $\pm\frac{3}{2}$ , and if y=0, x is $\pm\frac{3}{\sqrt{2}}$ . So the horizontal (y=0) axis is longer.

So you are correct. If the coefficient of $x^2$ is smaller, the major axis is horizontal, and if the coefficient of $y^2$ is smaller, the major axis is vertical.

- Hollywood

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