Another distance formula equation...did I do it right?

Question

This is another question related to the one I did earlier except I think I understand it now. I just need to know if I did it correctly.

Problem: Write an equation of the circle that has the given center and radius.

1. C (0,0); r = 6

My answers:
(x-0) + (y-0)= 6 or
(x-0)^2 = (y-0)^2 = 36

2. C(-2,5); r = 1/3
My answers:
(x-2) + (y-5) = 1/3 or
(x-2)^2 + (y-5)^2 = 1/9

Thanks for the help again!

Answer 1

You end up with the correct final answers (except that as mentioned, it should be (x+2)² in the second problem), but your intermediate step is atrocious. (x-x₀) + (y-y₀) = r does not imply that (x-x₀)² + (y-y₀)² = r², and in fact isn't even true: for the first one, (3√2, 3√2) lies on the circle (since (3√2)² + (3√2)² = 6²), but 3√2+3√2=6√2≠6. An accurate squaring of both sides would have yielded (x-x₀) + (y-y₀) = r → (x-x₀)² + (y-y₀)² + 2(x-x₀)(y-y₀)= r², which is definitely not the correct formula. Don't get lazy and start doing invalid steps and justify it to yourself by saying your errors cancel each other out, because you'll get into the habit of doing it and one day your errors won't cancel each other out, and then you'll get the wrong answer and not understand why.

Answer 2

The standard equation of a circle centered at the origin (0,0) is:

x^2 + y^2 = r^2

The radius given is 6.

You only need to plug in 6 for the radius and square it.

x^2 + y^2 = 36

+++++++++++++++++++++

When the circle has been moved away from the center, we use the equation:

(x - h)^2 +(y - k)^2 = r^2, where (h,k) is the center, r = the radius and (x,y) is any point on the circle.

We have this:

Point C(-2,5) = (h, k)

We also have r = 1/3

Let's plug and chug.

(x - h)^2 +(y - k)^2 = r^2

(x -(-2))^2 + (y - 5)^2 = (1/3)^2

(x + 2)^2 + (y - 5)^2 = 1/9

Guido

Answer 3

You're very close! The second one should be (x+2)^2, not (x-2)^2, since the center is at x = -2. Good job!

Answer 4

1) correct.

2) you forgot to put the opposite of -2. which is 2

(x-k)^2 + (y-h)^2 = r^2
(x - (-2))^2 + (y - 5)^2 = 1/3^2

(x + 2)^2 + (y-5)^2 = 1/9

Answer 5

1.) is perfect!

2.) the x-2 term should be x-(-2) = x+2

Answer 6

Looks good, nice job. On the second one it should be x+2 however