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Question 456674: conics 9x^2-36x-4y^2-36y-36=0 could you put this in standard form and show me how
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! conics 9x^2-36x-4y^2-36y-36=0 could you put this
in standard form and show me how.
Complete the square on the x terms and on the y terms:
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Rearrange:
9x^2 -36x -4y^2-36y = 36
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Complete the square:
9(x^2-4x+4) - 4(y^2+9y+4.5^2) = 36+36-4(4.5)^2
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9(x-2)^2 -4(y-4.5)^2 = -9
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-(x-2)^2 + (y-4.5)^2/(9/4) = 1
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Cheers,
Stan H.
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Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
The other tutor has a sign wrong.
9xē - 36x - 4yē - 36y - 36 = 0
Factor the coefficient of xē, which is 9, out of
the two x-terms
9(xē - 4x) - 4yē - 36y - 36 = 0
Factor the coefficient of yē, which is -4, out of
the two y-terms
9(xē - 4x) - 4(yē + 9y) - 36 = 0
Get the term -36 off the left side by adding
36 to both sides:
9(xē - 4x) - 4(yē + 9y) = 36
Multiply the coefficient of x inside the parentheses,
which is -4 by ― getting -2. Then square -2 getting +4.
Add +4 inside the first parentheses, and offset it by
adding +36 to the right side, since adding +4 inside
the first parentheses amounts to adding +36 to the left side
because of the 9 outside the first parentheses:
9(xē - 4x + 4) - 4(yē + 9y) = 36 + 36
Combine the 36 + 36 on the right as 72
9(xē - 4x + 4) - 4(yē + 9y) = 72
Multiply the coefficient of y inside the second parentheses,
which is +9 by ― getting +9/2. Then square +9/2 getting +81/4.
Add +81/4 inside the second parentheses, and offset it by
adding -81 to the right side, since adding +81/4 inside
the second parentheses amounts to adding -81 to the left side
because of the -4 outside the second parentheses:
9(xē - 4x + 4) - 4(yē + 9y + 81/4) = 72 - 81
Combine the 72 - 81 on the right as -9
9(xē - 4x + 4) - 4(yē + 9y + 81/4) = -9
Factor xē - 4x + 4 as (x - 2)(x - 2) and then as (x - 2)ē
Factor yē + 9y + 81/4 as (y + 9/2)(y + 9/2) and then as (y + 9/2)ē
9(x - 2)ē - 4(y + 9/2)ē = -9
Next we must get a 1 on the right.
So we divide all the terms by -9
9(x - 2)ē 4(y + 9/2)ē -9
- =
-9 -9 -9
(x - 2)ē 4(y + 9/2)ē
- + = 1
1 9
We must get the 4 off the top of the second term
by dividing top and bottom by 4
(x - 2)ē (y + 9/2)ē
- + = 1
1 9/4
Now reverse the two terms so that the positive term
comes first:
(y + 9/2)ē (x - 2)ē
+ = 1
9/4 1
Next we compare that to
(y - k)ē (x - 2)ē
- = 1
aē bē
which means that it is a hyperbola that opens upward
and downward
We see that h = 2, k = -9/2, aē = 9/4 so a = 3/2 and bē = 1 so b = 1
The center = (h,k) = (2,-9/2)
Edwin
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