Immediate help needed idenifing the conic

[arise1dwr]

X^2 - 4Y^2 - 4X -24Y = 48, IDENTIFY IF ITS A CIRCLE, PARABOLA, HYPERBOLA, OR ELLIPSE, SHOW WHY

 

[masters]

X^2 - 4Y^2 - 4X -24Y = 48, IDENTIFY IF ITS A CIRCLE, PARABOLA, HYPERBOLA, OR ELLIPSE, SHOW WHY

Hi aris1dwr,

\(\displaystyle x^2-4y^2-4x-24y=48\)

Group your x and y terms and complete the squares.

\(\displaystyle (x^2-4x+4)-4(y^2+6y+9)=48+4-36\)

\(\displaystyle (x-2)^2-4(y+3)^2=16\)

\(\displaystyle \frac{(x-2)^2}{16}-\frac{(y+3)^2}{4}=1\)

Now, what do you think?

Another way to tell is set the equation up in the form: \(\displaystyle Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)

(1) If A=0, or C=0, but not both, then it is a parabola.

(2) If A=C, then it is a circle.

(3) If A and C have the same sign and A is not equal to C, then it is an ellipse.

(4) If A and C have opposite signs, then it is a hyperbola.

Edit: Fixed typo.

 

[fasteddie65]

Master meant "A and C have opposite signs" not "opposite sines".

 

[masters]

Mastesr meant "A and C have opposite signs" not "opposite sines".

You are so right, my friend.