A circle is described by the equation x2 + y2 + 14x + 2y + 14 = 0. What are the coordinates for the center of the circle and the length of the radius?

Question

i need help

Answer 1

This can be solved by using the completing the square method. 

 

First, I am going to subtract the 14 from each side to get:

 

x²+y²+14x + 2y = -14

 

Now let's focus on the y's and then we can apply the same logic to the x's. For simplicity, I am going to put stuff with x variables aside, but we are going to figure out both coordinates in a bit.

 

y²+2y= -14

 

using the completing the square we must divide the coefficient in front of the y by 2, in this case, we use the 2y and then square it. 2/2 = 1, 1²= 1. add this 1 to both sides of the equation. 

 

 

y2+2y + 1= -14 + 1

 

This yields 
y2+2y + 1= -13. Now, we factor the left side of the equation. We get (y+1)*(y+1) = -13 aka (y+1)² = -13

 

ok we figured out the shift in the y axis for the center of the circle, now it is time to do it for the x's. I am going to keep the -13 in this case because when we use the completing the square method, we have to add to the right side of the equation when we solve for the x and y shift of the center.

x² + 14x = -13. Now we do the same method we did previously for y. 14/2 = 7, 7² = 49. let's add that 49 to each side of the eqn.

 

x² + 14x +49 = -13 + 49 This gives us:   x² + 14x +49 = 36. Factor out the left side of the equation like before, you get (x+7)² = 36

 

PUTTING IT ALTOGETHER:

(x+7)² + (y+1)² = 14 + 1 (from solve for y) + 49 (for solving for x). We get  (x+7)2 + (y+1)2 = 36. 

 

This equation is now in center-radius form. your center is -7,-1 and your radius is 6. 

If you are not sure how I got that last part, this is because the typical equation is formatted in this way:

(x-h)² + (y-k)² = r²

 

x,y is any point on the circle
h,k is the center (switch signs when moving from the actual coordinates to the eqn form.)

r is the radius

 

If you have any more questions let me know!




  

 

 

 

 

y2+2y + 1= -14 + 1

Answer 2

x²+ 14x + __+ y² + 2y +      = -14

You need to complete the square to fill in the two blanks.

14/2 = 7, 7² = 49     2/2 = 1, 1² = 1

x²+ 14x + 49 +y² + 2y + 1 = -14 +49 + 1 

(x+7)² + (y+1)² = 36

C(-7,-1)   r = 6