SOLUTION: What is the points of intersection of these equations x-2y-8=0 x^2=y^2-12x+6y+29=0

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Question 258000: What is the points of intersection of these equations
x-2y-8=0
x^2=y^2-12x+6y+29=0
Found 3 solutions by Greenfinch, Edwin McCravy, Alan3354:
Answer by Greenfinch(383)   (Show Source): You can put this solution on YOUR website!
Second equation is a circle, so reorganise it
(x^2 + 12 x) - (y^2 - 6y)+ 29 = 0 and then complete the square
(x^2 + 12x + 36) - ( y^2 - 6y + 9) - 45 + 29 = 0 which becomes
(x + 6)^2 - (y - 3)^2 = 4^2 which is a circle centre at -6, 3 and radius 4
other equation is y = (1/2)x - 4 or x = 2y + 8
Probably solve for y is easier
(2y + 14)^2 - (y - 3)^2 = 4^2
4y^2 + 56 y + 196 -(y^2 - 6y +9) = 16
3y^2 + 62y + 187 = 16
3y^2 + 62y + 171 = 0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=1792 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: -3.27799650382776, -17.3886701628389. Here's your graph:

Then use x = 2y + 8 to get the x figures
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!


Note: The other two tutors' solutions are both incorrect.




Solve the first equation for x




Substitute  for x in the second equation:













Factor:



Use zero-factor principle:

 gives solution 

 gives solution 

Now we must find the value of x for each of these
two values for y.

For  we substitute  for y in 









,)

or if you prefer, (x,y) = (8.4,.2)

For  we substitute  for y in 









So the other solution is (x,y) = (2,-3)

Edwin
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
The 2nd "equation" has too many equal signs.
If the 1st = is a +, then Edwin's solution is good.
If it's a minus sign, it's a different solution.
---------------
You can see in Greenfinch's photo that he's sound asleep.
Greenfinch said it's a circle, but used a minus sign which makes it a hyperbola.
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