A Circle Question.

**cromlix** · February 12th 2011, 06:51 AM

Hi,
Can anyone give me a hand with the following problem please?

For what range of values of k does: x^2+y^2+4kx-2ky-k-2+0
represent a circle?

I have tried using the radius formula :-

Sqrrt (-2k)^2 + (k)^2 -(k-1)

Sqrrt 4k^2 + k^2 -k+1

Sqrrt 5k^2 -k+1......
Thanks
Cromlix

**abhishekkgp** · February 12th 2011, 07:02 AM

Originally Posted by cromlix

Hi,
Can anyone give me a hand with the following problem please?

For what range of values of k does: x^2+y^2+4kx-2ky-k-2+0
represent a circle?

I have tried using the radius formula :-

Sqrrt (-2k)^2 + (k)^2 -(k-1)

Sqrrt 4k^2 + k^2 -k+1

Sqrrt 5k^2 -k+1......
Thanks
Cromlix

I guess the equation you wanted to write was $x^2+y^2+4kx-2ky-k-2=0$
the formula for the radius is $\sqrt{g^2+f^2-c}$
i guess you have made a calculation mistake there.

**Prove It** · February 12th 2011, 07:32 AM

Originally Posted by cromlix

Hi,
Can anyone give me a hand with the following problem please?

For what range of values of k does: x^2+y^2+4kx-2ky-k-2+0
represent a circle?

I have tried using the radius formula :-

Sqrrt (-2k)^2 + (k)^2 -(k-1)

Sqrrt 4k^2 + k^2 -k+1

Sqrrt 5k^2 -k+1......
Thanks
Cromlix

$\displaystyle x^2 + y^2 + 4k\,x - 2k\,y -k- 2 = 0$

$\displaystyle x^2 + 4k\,x + (2k)^2 + y^2 - 2k\,y + (-k)^2-k - 2 = (2k)^2 + (-k)^2$

$\displaystyle (x + 2k)^2 + (y - k)^2 -k- 2 = 4k^2 + k^2$

$\displaystyle (x + 2k)^2 + (y - k)^2 = 5k^2 + k+2$ .

Now all that's left is to show the values for $\displaystyle k$ for which the RHS is nonnegative (complete the square).

**Plato** · February 12th 2011, 07:48 AM

Although Prove_It missed a $k$ as in $(x+2k)^2+(y-k)^2=k+2+5k^2$ it does not change his answer.

**Prove It** · February 12th 2011, 07:54 AM

Originally Posted by Plato

Although Prove_It missed a $k$ as in $(x+2k)^2+(y-k)^2=k+2+5k^2$ it does not change his answer.

Fixed...

**cromlix** · February 14th 2011, 08:53 AM

Thanks for your work.
I have completed the square (x+2k)^2 + (y-k)^2= 5k^2+k+2

5( k^2 + 0.2k) +2
5( k^2 +0.2k +0.01) -0.05 + 2
5(k + 0.1)^2 +1.95

How do I find the range from this?

Thank you
Cromlix

**Prove It** · February 14th 2011, 01:19 PM

All you had to do was show that this is always nonnegative.

Can a square ever give a negative value?

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