SOLUTION: a conic section is described by x^2+6x+5/8y^2=-4. what type of conic section does this repersent?

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Question 632168: a conic section is described by x^2+6x+5/8y^2=-4. what type of conic section does this repersent?
Found 2 solutions by lwsshak3, Edwin McCravy:
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
a conic section is described by x^2+6x+5/8y^2=-4. what type of conic section does this repersent?
x^2+6x+5/8y^2=-4.
complete the square for the x-term
(x^2+6x+9)+5y^2/8=-4+9
(x+3)^2+5y^2=5
divide by 5
%28x%2B3%29%5E2%2F5%2By%5E2%2F1=1
This is an equation of an ellipse with horizontal major axis
Its standard form: %28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 a>b, (h,k)=(x,y) coordinates of center

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
There are two sets of rules and you can go by either one.  

The simpler one is for second degree polynomial equations which have no
xy term. It is a look-and-see rule that requires no calculation:

1. If there is an x² term but no y² term or a y² term but no x² term,
   the graph of the  equation is a parabola.

2. If the x² term and the y² term have equal coordinates when on the same
   side of the equal sign, the graph of the equation is a circle.

3. If the x² term and the y² term have the same sign when on the same
   side of the equal sign, the graph of the  equation is an ellipse.
   (Note than a circle is a special case of an ellipse when the x² and y²
    terms not only have the same sign, but also the same coefficient).

4. If the x² term and the y² term have opposite signs when on the same
   side of the equal sign, the graph of the equation is a hyperbola.
     
Your equation has no xy term and fits 3, so it is an ellipse.

----------------------------

There is a more general rule that applies to all second degree polynomials
regardless of whether they have an xy term or not.

------------------------------

The more general rule requires a calculation:

Put the equation in the form

Ax² + Bxy + Cy² + Dx + Ey + F = 0

then form the discriminant B²-4AC

If this is 0, the graph of the equation is a parabola.
If this is negative , the graph of the equation is an ellipse (or circle).
If this is positive , the graph of the equation is a hyperbola.


 x%5E2%2B6x%2Bexpr%285%2F8%29y%5E2=-4

1x² + 0xy + 5%2F8y² + 6x + 0y + 4 = 0

A=1  B=0  C=5%2F8  D=6  E=0  F=4

B²-4AC = 0²-4(1)(5%2F8} = -5%2F2

That is negative so the equation is of an ellipse by the general rule.

Edwin