With general form of circle, make it into standard form.

Question

Complete the square and write the equation of the circle into standard form.

5x^2-9x+5y^2-8y=0

Find the center (h,k) and radius r.

Answer 1

5( x^2 - 9/5x + ?) + 5(y^2 -8/5 y + ?) = 0

To complete the square first be sure it is only x^2, etc in the ( ).  Using the coefficient in front of x (and y), take 1/2 the value and then square the result.  That values times the coefficient outside the parentheses has to be added to the right side of the equation so the equation stays in balance.

5( x^2 - 9/5x + 81/100 ) + 5(y^2 -8/5 y + 64/100) = 0 + 405/100 + 320/100

5(x - 9/10)^2 + 5(y - 8/10)^2 = 725/100  DIVIDE ALL BY 5.

(x - 9/10)^2 + (y - 8/10)^2 = 725/500

 

Center of circle is (9/10, 8/10) and radius is the square root of 725/500 or 1.20.

 

Mike C.

Answer 2

Factor out a 5 and group equation by variable 

 

5(x² -9/5 X   ) +5(y² -8/5 y  )=0

 

take half of each middle term and square it and add to both side but remember you must multiply by 5 when adding to the right side. 

 

5(x2 -9/5 X +81/100 ) +5(y2 -8/5 y 64/100)=81/20+64/20

 

5(X-9/10)²  + 5(y-8/10)² =145/20

 

Divide everything by 5 and simplify

 

(X-9/10)² + (y-4/5)² =29/20

 

Center is (9/10,4/5) and radius is (sqrt(145))/10

 

 

Answer 3

Esther,

 

First, divide everything by 5 (the coefficient for the squared terms)

 

x² - (9/5)x + y² - (8/5)y = 0/5 = 0

 

Then, rewrite to look like:

 

(x² - (9/5)x +   ) + (y² - (8/5)y +   ) = 0

 

To complete the square for each quadratic, take half of each coefficient of the middle term for each quadratic and square it; e.g., half of -9/5 is -9/10, which squared it 81/100; half of -8/5 is -8/10 which squared it 64/100.

 

Now add the squares in the appropriate parentheses AND to the right side of the equation (don't forget to do this):

 

(x² - (9/5)x + 81/100) + (y² - (8/5)x + 64/100) = 81/100 + 64/100

 

Finally, factoring the two quadratics on the left side, we get:

 

(x - 9/10)² + (y - 8/10)² = 145/100

 

This is now in standard form:

 

(x - h)² + (y - k)² = r²

 

So, (h, k) = (9/10, 8/10 or 4/5) and r² = 145/100 so r = √(145/100) = 1.20

Answer 4

5 x ² -9 x + 5 y ² -8 y = 0

x ² -9/5 x +  y ² -8/5 y = 0

(x ² -9/5 x + 81/100 - 81/100) +  (y ² -8/5 y + 64/100 -64/100) = 0

 

(x -9/10)2 + (y - 8/10)2 = 81/100 + 64/100

(x  -9/10)²  + (y  - 8/10)² =  (12/10)²

 

the circle center is (9/10, 8/10), radius is 12/10