How do you know if the conic section $x^{2} - 9 y^{2} + 36 y - 45 = 0$ $x^2 -9y^2 +36y -45= 0$ is a parabola, an ellipse, a hyperbola, or a circle?

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Answer 1 · 2015-11-12T16:15:11

Look at the coefficients on $x^{2} and y^{2}$ $x^2 and y^2$

The coefficient on $x^{2} = 1$ $x^2 = 1$ and the coefficient on $y^{2} = - 9$ $y^2 = -9$ .

If the coefficients are different and also different signs (+ or -), then it must be hyperbola.

Detailed Explanation:

EVERY conic section can be written with the general equation :

$A x^{2} + B y^{2} + C x + D y + E = 0$ $Ax^2+By^2+Cx+Dy+E=0$

Here is how you distinguish the various conic sections from the coefficients in the general equation:

circle : $A = B$ $A=B$

ellipse : $A \neq B$ $AneB$ but A and B both have the SAME sign (+ or -)

hyperbola : $A \neq B$ $AneB$ but A and B both have DIFFERENT signs

parabola : either A or B equals 0 (only one squared term in the equation)

hope that helped

How do you know if the conic section x2−9y2+36y−45=0x^2 -9y^2 +36y -45= 0 is a parabola, an ellipse, a hyperbola, or a circle?