# Thread: Parabola Ellipse and Hyperbola

1. ## Parabola Ellipse and Hyperbola

How will u classify the following curves in terms of parabola, ellipse or hyperbola?

i) $\displaystyle 17x^2+12xy+8y^2-46x-28y+33=0$

ii) $\displaystyle x^2-5xy+y^2+8x-20y+15=0$

2. Originally Posted by varunnayudu
How will u classify the following curves in terms of parabola, ellipse or hyperbola?

i) $\displaystyle 17x^2+12xy+8y^2-46x-28y+33=0$

ii) $\displaystyle x^2-5xy+y^2+8x-20y+15=0$
Read this: Conic section - Wikipedia, the free encyclopedia

3. Hello, varunnayudu!

How will u classify the following curves as parabola, ellipse or hyperbola?

$\displaystyle 1)\;\;17x^2+12xy+8y^2-46x-28y+33\:=\:0$

$\displaystyle 2)\;\;x^2-5xy+y^2+8x-20y+15\:=\:0$
If you are assigned "rotated" conics, you should have been given more information.

We know the curve is rotated due to the appearance of the $\displaystyle xy$-term.

The general quadratic has the form: .$\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F \;=\;0$

. . It has a "discriminant": .$\displaystyle \Delta \:=\:B^2-4AC$ .
... look familiar?

. . . . . . . . $\displaystyle \begin{array}{|c|c|}\hline \Delta \:=\:0 & \text{parabola} \\ \hline \Delta \:<\:0 & \text{ellipse} \\ \hline \Delta \:>\:0 & \text{hyperbola}\\ \hline \end{array}$

4. ## can u give the answer

i have solved the two eq and got an answer but need to see if its right or wrong ...........Can u give what type of conic do the two eq form

5. ## what do u think.

Originally Posted by Soroban
Hello, varunnayudu!

If you are assigned "rotated" conics, you should have been given more information.

We know the curve is rotated due to the appearance of the $\displaystyle xy$-term.

The general quadratic has the form: .$\displaystyle Ax^2 + Bxy + Cy^2 + Dx + Ey + F \;=\;0$

. . It has a "discriminant": .$\displaystyle \Delta \:=\:B^2-4AC$ . ... look familiar?

. . . . . . . . $\displaystyle \begin{array}{|c|c|}\hline \Delta \:=\:0 & \text{parabola} \\ \hline \Delta \:<\:0 & \text{ellipse} \\ \hline \Delta \:>\:0 & \text{hyperbola}\\ \hline \end{array}$

I have done it this way .Is this right.........

i) $\displaystyle 17x^2 \ + \ 12xy \ + \ 8 y^2 \ - \ 46x \ - \ 28y \ + 33 \ = \ 0$

$\displaystyle a \ = \ 17 ; \ b \ = \ 8 ; h \ = \ 6;$

$\displaystyle ab-h^2 \ = \ 136 - 36 = \ 100$

since $\displaystyle ab-h^2 > 0$ therefore the eq is an hyperbola.

ii) $\displaystyle x^2 - 5xy + y^2 + 8x - 20y +15 = 0$

$\displaystyle a=1 ; \ b = 1; \ h = - \frac{5}{2}$

$\displaystyle \therefore \ ab-h^2 = 1 - \frac{25}{4} = \frac{4 - 25}{4} = - \frac{21}{4} \ = -5.25 \ < \ 0$

$\displaystyle \therefore since \ ab-h^2 < 0$ it is an ellipse.