SOLUTION: Please help me, I don't understand conic intersection at all. x^2 - 4y^2 = 16 y = 3x - 3

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Question 553901: Please help me, I don't understand conic intersection at all.
x^2 - 4y^2 = 16
y = 3x - 3
Found 2 solutions by Alan3354, AnlytcPhil:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
x^2 - 4y^2 = 16
y = 3x - 3
-----------
Sub for y in the 1st equation.
Find x & y, forget about "conic intersection."

Answer by AnlytcPhil(1810) About Me  (Show Source):
You can put this solution on YOUR website!
This problem has no real solutions. Let me explain what is 
going on here.  

The equation 

x² - 4y² = 16

if you were to solve it for y and plot a bunch of points, like these:

(-8.5,-3.75), (-8.5,3.75), (-5,-1.5), (-5,1.5), (-4,0)
(4,0), (5,-1.5), (5,1.5), (8.5,-3.75), (8.5,3.75)

you would get this graph, called a hyperbola.  It is called a "conic section"
because there are several kinds of curves that can be gotten by slicing a
cone different ways.  The other tutor is right that you need not be
concered about that at this time.  Anyway here is the graph of x² - 4y² = 16.



Now let's look at the other equation, 

y = 3x - 3

If you were to plot some points, like these:
 
(0,-3), (1,0), (2,3)

you will get a straight line like thisL



But if you put them both on the same graph, like this:



you find that they don't intersect at all!  That means there
cannot be any REAL point where they intersect. 

Suppose you did as the other tutor said:

x² - 4y² = 16
y = 3x - 3

Substitute 3x - 3 for y in the first equation:

x² - 4(3x - 3)² = 16
x² - 4(3x - 3)(3x - 3) = 16
x² - 4(9x² - 18x + 9) = 16
x² - 36x² + 72x - 36 = 16
-35x² + 72x - 36 = 16
-35x² + 72x - 52 = 0
 35x² - 72x + 52 = 0

That does not factor and if you use the quadratic formula:

x = %28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+  
 
x =  

x = %2872+%2B-+sqrt%285184-7280+%29%29%2F%2870%29+ 

x = %2872+%2B-+sqrt%28-2096%29%29%2F70+

x = %2872+%2B-+i%2Asqrt%282096%29%29%2F70+

x = %2872+%2B-+i%2Asqrt%2816%2A131%29%29%2F70+

x = %2872+%2B-+4i%2Asqrt%28131%29%29%2F70+

x = %284%2818+%2B-+i%2Asqrt%28131%29%29%29%2F70+

x = %282%2818+%2B-+i%2Asqrt%28131%29%29%29%2F35+

Which is an imaginary answer, not a real answer

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

Now if the second equation had been 10y + 3x = 0
instead and your problem had been:

x² - 4y² = 16
10y + 3x = 0

and we found and plotted these points, say

(0,0), (4, -1.2), (7,-2.1) 

and had gotten this line instead, there would have
been two real solutions at the points where these two
graphs intersect, as shown with the two small circles. 



Suppose you did as the other tutor said in this
problem:

x² - 4y² = 16
10y = 3x

Solve the second equation for y

y = 3x%2F10

Substitute 3x%2F10 for y in the first equation:
 
x² - 4y² = 16

x² - 4%283x%2F10%29%5E2 = 16

x² - 4%289x%5E2%2F100%29 = 16

x² - %289x%5E2%2F25%29 = 16

Multiply through by 25 to clear of fractions:

25x² - 9x² = 400
      16x² = 400
        x² = 25
         x = ±5

Using x = +5 you would have gotten

y = 3x%2F10
y = 3%285%29%2F10%7D%7D%0D%0Ay+=+%7B%7B%7B15%2F10
y = 3%2F2

So one solution would be the point (5,3%2F2) or (5,1.5)

Using x = -5 you would have gotten

y = 3x%2F10
y = 3%28-5%29%2F10%7D%7D%0D%0Ay+=+%7B%7B%7B-15%2F10
y = -3%2F2

So the other solution would be (-5,-3%2F2) or (-5,-1.5)

Notice that the points where the graphs intersect are indeed
those very points (5,1.5) and (-5,-1.5)



Edwin