My favorite method

[The Highlander]

I believe that "Completing the Square" is not the way you are expected to approach this problem.

At least that is the way it is taught here in Scotland (and, I believe, throughout the rest of the UK too; other countries may differ in their approaches, of course).

 

[Deleted member 4993]

Despite what others have said, I believe that "Completing the Square" is not the way you are expected to approach this problem.

At least that is the way it is taught here in Scotland (and, I believe, throughout the rest of the UK too; other countries may differ in their approaches, of course).

I disagree -

"completing the square" is a more general method,

applicable to general "form", and this is preferable method in the USA and India (in two educational system that I am familiar with). This method is applicable for any conic section, without any significant change.

 

[The Highlander]

I disagree -

"completing the square" is a more general method,

applicable to general "form", and this is preferable method in the USA and India (in two educational system that I am familiar with).

You are, of course, perfectly entitled to "disagree" and, as I said in my post, "other countries may differ in their approaches" but here (in the UK) we specifically teach the "Expanded Form" \(\displaystyle \left(x^2+y^2+2gx+2fy+c=0\right)\) where the centre is \(\displaystyle \left(-g,-f\right)\) and the radius, \(\displaystyle r=\sqrt{g^2+f^2-c}\). and, when an equation is given in that form, we would expect the student to use those facts to determine the centre & radius of the circle.

I have to say that, despite accepting your comment that "'completing the square' is a more general method", in this particular case, it seems to me that finding the answers required in the OP's problem is simpler (& quicker) using the method I suggest; I can read off the centre's coordinates directly from the equation provided and writing down three numbers to plug into the formula (using the correct signs, ofc ?) gives me the radius in seconds! Thus leaving me free to spend the remainder of my time to sketch the circle using those parameters. ?

Completing the Square (for me at least) would involve writing down the original equation followed by several lines of algebraic manipulation to get the answers required.

Of course, I (we?) have no idea where the OP is based or what s/he has been taught so maybe s/he is, indeed, expected to follow your "more general method"; I was just trying to provide what I thought was the "best" way to tackle the problem as originally posted.

 

[Steven G]

@The Highlander,
It is great to see how the same problem can be taught different ways and both are quite efficient. I agree that your form has its advantages.

 

[mmm4444bot]

I was just trying to provide what I thought was the "best" way

We don't mind tutors acknowledging a suggested approach as their personal favorite, but you could have phrased it better.

… you are expected to [use this] simple formula: sqrt(g^2 + f^2 + c).

I believe that "Completing the Square" is not the way you are expected to approach this problem.

Emphasized in bold, and italics.
[imath]\;[/imath]

 

[Cubist]

@The Highlander FYI it seems that a major exam board in England ??????? teaches the complete the square method...


Taken from the AQA A level mathematics 2017 specification

However, I honestly don't remember the method that I was taught when I took the exam myself such a very long time ago. (I can only just about remember what I had for breakfast today.) I think my exam board was JMB which no longer exists

 

[topsquark]

@The Highlander: As a Physicist I believe in solving the problem in any way that you have available. (Unfortunately for me and my students I almost invariably choose the hard way!) I, too, feel that completing the square is the way to solve this one, but that's how I was taught. Other methods are equally valid and I thank you for posting yours.

-Dan

 

[lookagain]

... but here (in the UK) we specifically teach the "Expanded Form" \(\displaystyle \left(x^2+y^2+2gx+2fy+c=0\right)\) where the centre is \(\displaystyle \left(-g,-f\right)\) and the radius, \(\displaystyle r=\sqrt{g^2+f^2-c}\). and, ...

This part in the quote box sounds as if in the United Kingdom they exclusively teach that "expanded form" that you wrote out.
That is not true as evidenced by https://www.sqa.org.uk/files_ccc/MathematicsPaper2SQPH.pdf