MHB What caused the wrong answer for determining the center and radius of Circle 2?

[mathdad]

Determine the center and radius of circle.

View attachment 7463

 

[Joppy]

Why not continue and find out?

Doing so will give you experience, and experience will allow you to answer these sorts of question for yourself in future situations.

 

[mathdad]

3x^2 + 3y^2 + 5x - 4y = 1

3x^2 + 5x + 3y^2 - 4y = 1

x^2 + (5/3)x + y^2 -(4/3)y = 1/4

Half of (5/3) is (5/6). Then (5/6)^2 = (25/36).

Half of -(4/3) is -(2/3). Then (-2/3)^2 = (4/9).

We add (25/36) and (4/9) on both sides of the equation.

x^2 + (5/3)x + (25/36) + y^2 -(4/3)y + (4/9) = (1/4) + (25/36) + (4/9)

Factor left side and calculate the right side.

(x + 5/6)(x + 5/6) + (y - 2/3)(y - 2/3) = (25/18)

(x + 5/6)^2 + (y - 2/3)^2 = (25/18)

The center is (h, k) = (-5/6, 2/3).

Let r = radius

r^2 = (25/18)

sqrt{r^2} = sqrt{25/18}

r = [5•sqrt{2}]/6

Is this correct?

 

[HallsofIvy]

3x^2 + 3y^2 + 5x - 4y = 1

3x^2 + 5x + 3y^2 - 4y = 1

x^2 + (5/3)x + y^2 -(4/3)y = 1/4

= 1/3, not 1/4.

Half of (5/3) is (5/6). Then (5/6)^2 = (25/36).

Half of -(4/3) is -(2/3). Then (-2/3)^2 = (4/9).

We add (25/36) and (4/9) on both sides of the equation.

x^2 + (5/3)x + (25/36) + y^2 -(4/3)y + (4/9) = (1/4) + (25/36) + (4/9)

Again, the right side should be 1/3+ 25/36+ 4/9. 1/3, not 1/4.

Factor left side and calculate the right side.

(x + 5/6)(x + 5/6) + (y - 2/3)(y - 2/3) = (25/18)

(x + 5/6)^2 + (y - 2/3)^2 = (25/18)

1/3+ 25/36+ 4/9= 12/36+ 25/36+ 16/36= 53/36

The center is (h, k) = (-5/6, 2/3).

Let r = radius

r^2 = (25/18)

sqrt{r^2} = sqrt{25/18}

r = [5•sqrt{2}]/6

Is this correct?

Not the radius. The radius is sqrt(53}/6.

 

[mathdad]

On the right side, I should have 1/3, as you said, not 1/4. Simple computation error that led to the wrong answer. Overall, I understand the procedure which is more important.