SOLUTION: Please help Transform to standard form and describe the graph of each equation. 1. 4x²+1=-4y²+4y 2. 36x²+36y²+36y=48x+199 3. 9x²+9y²+6x=-5 4. 3x²-5y=24-3y²-40 5. x²+y²+3x-5y+1

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Please help Transform to standard form and describe the graph of each equation. 1. 4x²+1=-4y²+4y 2. 36x²+36y²+36y=48x+199 3. 9x²+9y²+6x=-5 4. 3x²-5y=24-3y²-40 5. x²+y²+3x-5y+1      Log On


   

Question 718034: Please help Transform to standard form and describe the graph of each equation.
1. 4x²+1=-4y²+4y
2. 36x²+36y²+36y=48x+199
3. 9x²+9y²+6x=-5
4. 3x²-5y=24-3y²-40
5. x²+y²+3x-5y+17/2=0
thank you
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Those are all equations of circles and they are all done the same way,

I'll do the 2nd one as it is the most complete and the 3rd one is weird:

2. 36x²+36y²+36y=48x+199

Get the x-term next to the x² term and the y-term next to the y² term.
Get only the constant term on the right.

36x²-48x + 36y² + 36y = 199 

Divide every term through by the common coefficient of the x²
and the y² terms, which is 36

x² - 48%2F36x + y² + y = 199%2F36

Reduce the fraction 48%2F36 to 4%2F3

x² - 4%2F3x + y² + y = 199%2F36

Complete the squares on the lft:

1. Multiply the coefficient of x, which is -4%2F3, by 1%2F2,
getting -2%2F3.
2. Square this result, getting %28-2%2F3%29%5E2 = 4%2F9
3. Add this right after the x term on the left side and also 
add it to the right side.

1. Multiply the coefficient of y, which is 1, by 1%2F2,
getting 1%2F2.
2. Square this result, getting %281%2F2%29%5E2 = 1%2F4
3. Add this right after the y term on the left side and also 
add it to the right side.

x² - 4%2F3x + 4%2F9 + y² + y + 1%2F4 = 199%2F36 + 4%2F9 + 1%2F4

Factor the first three terms as (x - 2%2F3)(x - 2%2F3) which can be
written as the square (x - 2%2F3)².

Factor the last three terms on the left as (y - 1%2F2)(y - 1%2F2) which can be 
written as the square (y - 1%2F2)².

Combine the terms on the right by getting an LCD of 36
199%2F36 + 4%2F9 + 1%2F4 = 199%2F36 + 16%2F36 + 9%2F36 = %28199%2B16%2B9%29%2F36 = 224%2F36 = 56%2F9

(x - 2%2F3)² + (y - 1%2F2)² = 56%2F9

That is a circle with center (2%2F3,1%2F2) and radius expr%284%2F3%29sqrt%2814%29.

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

9x²+9y²+6x=-5

Get the x-term next to the x² term and the y-term next to the y² term.
The constant term is already on the right.

9x² + 6x + 9y² = -5

Divide every term through by the common coefficient of the x²
and the y² terms, which is 9:

x² - 6%2F9x + y² = -5%2F9

Reduce the fraction 6%2F9 to 2%2F3

x² - 2%2F3x + y² = -5%2F9

Complete the squares on the left:

1. Multiply the coefficient of x, which is -2%2F3, by 1%2F2,
getting -1%2F3.
2. Square this result, getting %28-1%2F3%29%5E2 = 1%2F9
3. Add this right after the x term on the left side and also 
add it to the right side.

Since there is no y term, write y² as (y - 0)²

x² - 4%2F3x + 1%2F9 + (y - 0)² = -5%2F9 + 1%2F9

Factor the first three terms as (x - 1%2F3)(x - 1%2F3) which can be
written as the square (x - 1%2F3)².

Combine the terms on the right:
-5%2F9 + 1%2F9 = -4%2F9

(x - 1%2F9)² + (y - 0)² = -4%2F9

That is an imaginary circle with a real center of (1%2F9,0) and an imaginary
radius of 2%2F3i.  Really weird.  Was that -5 on the right supposed to be 5?


Edwin

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