SOLUTION: what is the graph of the equation 6x^2-24x-5y^2-10y-11=0? How do yo identify the solutions of the system of equations y^2-x^2=16

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: what is the graph of the equation 6x^2-24x-5y^2-10y-11=0? How do yo identify the solutions of the system of equations y^2-x^2=16       Log On


   

Question 223022: what is the graph of the equation 6x^2-24x-5y^2-10y-11=0?
How do yo identify the solutions of the system of equations y^2-x^2=16
x^2-y^2=16
*Can you please show steps so i will know how to solve these kinds of problems in the future
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

From Wikipedia (see conic section article), if you have a conic of the form Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0, then one of the following is possible:


1) If B%5E2+-+4AC+%3C+0, then the equation represents an ellipse


Special Case: if A+=+C and B+=+0, the equation represents a circle (which is a special case of an ellipse).


or...


2) If B%5E2+-+4AC+=+0, then the equation represents a parabola.


or...


3) if B%5E2+-+4AC+%3E+0, then the equation represents a hyperbola.


Special Case: if A%2BC=0 (ie A=-C), then the conic is a rectangular hyperbola.


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


Now let's solve the problem at hand.


6x%5E2-24x-5y%5E2-10y-11=0 Start with the given conic.


6x%5E2-5y%5E2-24x-10y-11=0 Rearrange the conic into the form Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0.


By looking at the last equation, we see that A=6, B=0, C=-5, D=-24, E=-10 and F=-11(just match up the corresponding coefficients).


So in order to classify the given conic, we're going to use the formula B%5E2-4AC and compare it to one of the cases described above.


B%5E2-4AC Start with the given formula.


%280%29%5E2-4%286%29%28-5%29 Plug in A=6, B=0, and C=-5.


0-4%286%29%28-5%29 Square 0 to get 0.


0--120 Multiply to get 4%286%29%28-5%29 -120.


120 Combine like terms.


So B%5E2-4AC=120.


Since B%5E2-4AC%3E0, this means that the conic is an hyperbola.


Here's a graph to verify that result:


Graph of 6x%5E2-24x-5y%5E2-10y-11=0


======================================================================
# 2

y%5E2-x%5E2=16 Start with the first equation


y%5E2=x%5E2%2B16 Add x%5E2 to both sides.

--------

x%5E2-y%5E2=16 Move onto the second equation.


x%5E2-%28x%5E2%2B16%29=16 Plug in y%5E2=x%5E2%2B16


x%5E2-x%5E2-16=16 Distribute


-16=16 Combine like terms.


Since the equation is NEVER true for any value of 'x' or 'y', this means that there are no solutions to the system. Visually, this means that the two graphs will NEVER intersect (ie cross).