The circle with the equation x^2+y^2-2x+4y=0

[Jaco Viljoen]

Mod note: Moved from technical math section, so no template.
Hi All, i couldn't find any geometry under coursework and homework, but saw some geometry under general math.
I hope this is the right place, please advise and move if I have posted in the wrong place.The circle with equation x²+y²-2x+4y=0 has a centre C and radius r where
1. C(1,-2); r=Sqrt 5
2. C(-1,2); r=Sqrt 5
3. C(1,-4); r=sqrt 17
4. C(-1,4); r=17
5. C(1,-2); r=5

I have tried a couple of things:
Simplifying the equation:
x²+y²-2x+4y=0
x(x-2)+y(y+4)
x=2 and y=-4 This doesn't correspond to any of my options so can't be correct?

Using the standard form of the equation of a circle:

1.
(x-h)²+(y-k)²=r²
(x-1)²+(y+2)²=r²
x²-x-x+1+y²+2y+2y+4=r²
x²-2x+1+y²+4y+4=r
x²+y²-2x+4y+5=r²
x²+y²-2x+4y=-5
radius can't be negative...
so r=sqrt5

2.
(x-h)²+(y-k)²=r²
(x+1)²+(y-2)²=r²
(x+1)(x+1)+(y-2)(y-2)=r²
x²+1x+1x+1+y²-2y-2y-4=r²
x²+2x+y²-4y-3=r²
x²+y²+2x-4y=sqrt3
This doesn't correspond to any of my options so can't be correct.

3.
(x-h)²+(y-k)²=r²
(x-1)²+(y-(-4))²=2²
(x-1)(x-1)+(y-2)(y-2)=r²
x²-1x-1x+1+y²-2y-2y+4=r
x²+y²-2x-4y+5=r²
doesn't match the circle's equation

4.
(x-h)²+(y-k)²=r²
(x+1)²+(y-4)²=r²
(x+1)(x+1)+(y-4)(y-4)=r²
x²+1x+1x+1+y²-4y-4y+8=r²
x²+y²+2x-8y+9=r²
doesn't match the circle's equation

5.
(x-h)²+(y-k)²=r²
(x-1)²+(y+2)²=r²
x²-x-x+1+y²+2y+2y+4=r²
x²-2x+1+y²+4y+4=r
x²+y²-2x+4y+5=r²

I would really appreciate some direction on this.
Thank you all.

 

[Nick Papachrist]

Hello Jaco,
when you have a circle equation in form of x²+y²+Ax+By+C=0 in order to find the radius you should use the equation r=√(A² + B² -4C) / 2
and for the center of the circle K( -A / 2 , - B / 2 ) if you replace the variables with the numbers you should get the 1st answer.

 

[Jaco Viljoen]

Thank you Nick,
I do not know these formulas, or just don't recognise them.(where can I find these formulas? If you don't mind?)

Is what I have done here correct?
1.
(x-h)2+(y-k)2=r2
(x-1)2+(y+2)2=r2
x2-x-x+1+y2+2y+2y+4=r2
x2-2x+1+y2+4y+4=r
x2+y2-2x+4y+5=r2
x2+y2-2x+4y=-5
radius can't be negative...
so r=sqrt5

Thank you again.
Jaco

 

[Mentallic]

Jaco, don't use the formulae that Nick mentioned. The answers can more easily be found by completing the square similarly to what you've been trying to do.

There are two kinds of problems of these types: either you're given the details of the circle, such as it has centre (h,k) and radius r, which then means the equation for the circle is
(x-h)^2+(y-k)^2=r^2

or you're expected to go in the opposite direction, which is to be given an equation and find the details of the circle from that. This is always the more interesting one because they'll never give you the equation in the form above, so you have to manipulate it into such a form to get the answers.

So you're given
x^2+y^2-2x+4y=0

Looking just as the x terms, on the left hand side (LHS) we have x^2-2x. Now, there is only one value of h in (x-h)^2 that gives us that result (ignoring the constants), and that is of course h=1, giving us
(x-1)^2=x^2-2x+1
If we chose any other h then we'd have a different coefficient of x and would thus have the wrong answer. But what about the +1? Well, 1-1=0, right? So let's do just that to make the two expressions equal.

We have
x^2-2x
but want to complete the square on it, and we know that
(x-1)^2=x^2-2x+1
and so
x^2-2x=x^2-2x+1-1=(x^2-2x+1)-1=(x-1)^2-1

Now do the same for the y values.

 

[Nick Papachrist]

A proof of the previous formulas I used :
Set K( Xo , Yo ) , a point where the center of the circle is located. Then from the classical circle equation : (X - X_o)² + (Y - Y_o)² = r² ((1)) if you expand this equation you will get
X² - 2X*X_o + X_o² + Y² - 2Y*Y_o + Y_o² = r² ((2)) now if you set A = -2X_o and B = -2Y_o and C=X_o²+Y_o²-r² from the equation ((2)) and bring r² to the other side you will get A²+B²-4C = (-2X_o)² + (-2Y_o)² -4(X_o² + Y_o² - r²) = 4X_o² + 4Y_o² -4X_o² -4Y_o² +4r² = 4r² > 0 ((3)) now from equation ((3)) if you solve for the radius you will get the formula : r = √( A²+ B² -4C ) / 2 . Now to find the coordinates of the center K, for ever circle in form of x²+ y²+ Ax + By + C = 0 <=> x² + 2*(A/2)*x + A²/4 + y²+ 2*(B/2)*y + B²/4 = A²/4 + B²/4 - C if you factorize you will end up in the equation : ( x+A/2 )² + ( y+B/2 )² = (A²+B² - 4C)/4 ((4)) from equation ((3)) you can simply observe that A²+B² - 4C = 4r² and if you replace it to the equation ((4)) you will see the classical circle equation. So the coordinates from equation ((4)) are K( -A/2 , -B/2 ) .

Now for x²+y²-2x+4y=0 you can solve in another way. You can add both sides with 5 and you 'll get : x²+ y²- 2x + 4y + 5 = 5 and by factorizing you will end up in the form
(x-1)²+(y+2)² = 5 but r²=5 so r = √5, so the coordinates are K(1,-2) and r=√5 . Generally all you need is to factorize it to the form of (X - X_o)²+( Y - Y_o )² = r² to find both radius and coordinates of the center without involving the equations i showed above.

Thanks for replying Jaco

 

[Jaco Viljoen]

Mentallic,
I am not following,
(x−1)²=x²−2x+1
I don't know why this was done, but would y be:
y²+4y
(y-(-2))²
(y+2)²=y²+4y+4
(y+2)²-4

 

[Nick Papachrist]

Jaco if you multiply (x-1)(x-1) you will get x²-2x + 1

 

[Jaco Viljoen]

Jaco if you multiply (x-1)(x-1) you will get x²-2x + 1

I understand this, but why have we removed the y terms? I have never seen or done this...

 

[Nick Papachrist]

he didnt removed it he just isolated it from the equation to show you that you need to complete the square of x² + 2x by adding 1 to both sides same for y check my answer at the end .

"Now for x²+y²-2x+4y=0 you can solve in another way. You can add both sides with 5 and you 'll get : x²+ y²- 2x + 4y + 5 = 5 and by factorizing you will end up in the form
(x-1)²+(y+2)²= 5 but r²=5 so r = √5, so the coordinates are K(1,-2) and r=√5 . Generally all you need is to factorize it to the form of (X - Xo)²+( Y - Yo )²= r² to find both radius and coordinates of the center without involving the equations i showed above. "

 

[Jaco Viljoen]

Shucks, I am still feeling lost, I am going to see if I can find some more info on this and will try this again,
Thank you for the help, Nick and Mentallic.

I will be back on this thread as soon as I understand why these steps were taken.
Thank you again.

Jaco

 

[Mentallic]

I understand this, but why have we removed the y terms? I have never seen or done this...

I haven't, I've just focused on the x terms for the moment.
Let's focus on the LHS of the equation:

x^2+y^2-2x+4y
=x^2-2x+y^2+4y
=x^2-2x+1-1+y^2+4y+4-4
=(x-1)^2-1+(y+2)^2-4

 

[Jaco Viljoen]

Oh, I think I get it now.

x^2+y^2-2x+4y
=x^2-2x+y^2+4y
=x^2-2x+1-1+y^2+4y+4-4 , add half of x-term^2 and add half of y-term^2
=(x-1)^2-1+(y+2)^2-4
=(x-1)^2+(y+2)^2=+1+4
x=1, y=-2, r=√5

 

[Mentallic]

Very good :)

And just one final thing. If you're unsure about what to add and subtract, complet the square first and then find what the constant is from that. For example, given

x^2+10x

we know that to complete the square, we take half of the coefficient of x (5 in this case) and plug it in for h in (x+h)², giving us
(x+5)^2
and then if we expand this expression, the constant turns out to be 25, hence we have to add and subtract 25.

 

[SammyS]

Just focus on the following part of your post:

...
=(x-1)^2-1+(y+2)^2-4
\color{red}{=}(x-1)^2+(y+2)^2=+1+4

That leading "=" sign should not be in the last line.
$\displaystyle\ (x-1)^2-1+(y+2)^2-4 \ne (x-1)^2+(y+2)^2\$

 

[Jaco Viljoen]

Thank you for pointing this out Sammy,