Math's matching parameters!

[Ryan$]

Hi guys ! so sorry once again to post here like this, and I'm not saying it's wrong , I just want o think over it in other aspects which can help me to understand why it's right !
I'm truly confused on this:
the teacher said that we have (X-4)^2+(Y-5)^2=25 then he said it's circle ! here I'm with him, them immediately he said that :
A=4
B=5
R=5
which (X-A)^2+(Y-B)^2=R^2
without even thinking or solving the equation .. how did he do that? and why we can immediately say that
A=4
B=5
R=5
? is there a method or rule in math called matching parameters?! I doubt he matched parameters immediately ..why it's right?! I mean is in math there's what called matching's approach/method that I can use it without solving the equation for finding the parameters A,B,R?!

 

[lev888]

The solution of that equation is a set of points. They form a circle. That's why this equation is called equation of a circle. Is this your current topic? Did you read the textbook?

 

[pka]

Hi guys ! so sorry once again to post here like this, and I'm not saying it's wrong , I just want o think over it in other aspects which can help me to understand why it's right !
I'm truly confused on this:
the teacher said that we have (X-4)^2+(Y-5)^2=25 then he said it's circle ! here I'm with him, them immediately he said that :
A=4
B=5
R=5
which (X-A)^2+(Y-B)^2=R^2
without even thinking or solving the equation .. how did he do that? and why we can immediately say that

This is an example of your not knowing the prerequisites required to do the material.
\(\displaystyle (x-h)^2+(y-k)^2=r^2\) is a circle with center \(\displaystyle (h,k)\) and radius \(\displaystyle |r|\).
\(\displaystyle (X-4)^2+(Y-5)^2=25\) is a circle with center \(\displaystyle (4,5)\) and radius \(\displaystyle 5\).
\(\displaystyle (X+8)^2+Y^2=10\) is a circle with center \(\displaystyle (-8,0)\) and radius \(\displaystyle \sqrt{10}\).

\(\displaystyle x^2+6x+y^2-4y-3=0\\(x^2+6x+9)+(y^2-4y+4)=9+4+3\\(x+3)^2+(y-2)^2=16\) is a circle with center \(\displaystyle (-3,2)\) and radius \(\displaystyle 4\).

 

[ksdhart2]

Perhaps what's confusing you is that use of variables in this problem. Let's instead use placeholders and see what we have. The generic equation of a circle is:

\(\displaystyle (X - {\color{red}\boxed{\: \:}})^2 + (Y - {\color{blue}\boxed{\: \:}})^2 = {\color{limegreen}\boxed{\: \:}}^2\)

Your instructor wrote an equation for a specific circle. Let's see if we can't match up those terms (note that \(5^2 = 25\)):

\(\displaystyle (X - {\color{red}\boxed{\color{black} 4}})^2 + (Y - {\color{blue}\boxed{\color{black} 5}})^2 = {\color{limegreen}\boxed{\color{black} 5}}^2\)

Here, all we did was write 4 in the red box, 5 in the blue box, and 5 in the green box. Suppose instead that we wrote the general equation of a circle as:

\(\displaystyle (X - {\color{red}A})^2 + (Y - {\color{blue}B})^2 = {\color{limegreen}R}^2\)

From this, three things should be evident:

\(\displaystyle {\color{red} A} = {\color{red}\boxed{\: \:}}= {\color{red}\text{???}}\)

\(\displaystyle {\color{blue} B} = {\color{blue}\boxed{\: \:}}= {\color{blue}\text{???}}\)

\(\displaystyle {\color{limegreen} R} = {\color{limegreen}\boxed{\: \:}}= {\color{limegreen}\text{???}}\)

 

[LCKurtz]

You should recognize the distance formula. The distance from [MATH](x,y)[/MATH] to [MATH](a,b)[/MATH] is [MATH]\sqrt{(x-a)^2+(y-b)^2}[/MATH]. Set that equal to [MATH]r[/MATH], square both sides, and you have the standard equation of a circle with center [MATH](a,b)[/MATH] and radius [MATH]r[/MATH].

 

[mmm4444bot]

… the teacher said … (X-4)^2+(Y-5)^2=25 … them immediately … said … A=4 B=5 R=5 without even thinking …

You're wrong. The teacher did think, and they immediately recognized the algebraic pattern previously studied and learned.

… I doubt he matched parameters immediately …

Yes, that's exactly what he did, and you could do it too, if you were to study (and practice) algebra from the beginning -- versus jumping in the middle of a lake before learning how to swim!

To be frank, you're watching the wrong videos and you're not doing the work.

\(\;\)

 

[pka]

This is an example of your not knowing the prerequisites required to do the material.
\(\displaystyle (x-h)^2+(y-k)^2=r^2\) is a circle with center \(\displaystyle (h,k)\) and radius \(\displaystyle |r|\).
\(\displaystyle (X-4)^2+(Y-5)^2=25\) is a circle with center \(\displaystyle (4,5)\) and radius \(\displaystyle 5\).
\(\displaystyle (X+8)^2+Y^2=10\) is a circle with center \(\displaystyle (-8,0)\) and radius \(\displaystyle \sqrt{10}\).

\(\displaystyle x^2+6x+y^2-4y-3=0\\(x^2+6x+9)+(y^2-4y+4)=9+4+3\\(x+3)^2+(y-2)^2=16\) is a circle with center \(\displaystyle (-3,2)\) and radius \(\displaystyle 4\).

Look at this link.

 

[JeffM]

Hi guys ! so sorry once again to post here like this, and I'm not saying it's wrong , I just want o think over it in other aspects which can help me to understand why it's right !
I'm truly confused on this:
the teacher said that we have (X-4)^2+(Y-5)^2=25 then he said it's circle ! here I'm with him, them immediately he said that :
A=4
B=5
R=5
which (X-A)^2+(Y-B)^2=R^2
without even thinking or solving the equation .. how did he do that? and why we can immediately say that
A=4
B=5
R=5
? is there a method or rule in math called matching parameters?! I doubt he matched parameters immediately ..why it's right?! I mean is in math there's what called matching's approach/method that I can use it without solving the equation for finding the parameters A,B,R?!

You failed to say what A, B, and R are! Moreover, what do you understand by a parameter?

You keep treating math as if it is some sort of magic. A, B, and R must be defined for something to make sense. If A means the number of states in the U.S.A., then A = 4 is the WRONG answer, no matter what your teacher says. Pay attention to what letters represent.

In this case, A stands for the x-coordinate of the center of a circle. B stands for the y-coordinate of the center of that same circle. And R stands for the length of the radius of that circle.

There is a general formula that say
[MATH](x - h)^2 + (y - k)^2 = r^2 \text { is the equation of the circle with center at } (h,\ k) \text { and radius of r}.[/MATH]...edited typo

That is a general truth. So you can apply it to any specific case.

By the way, thank you VERY MUCH for giving a specific example.

 

[Ryan$]

thanks alot guys ! helped me so much

 

[Ryan$]

Hi guys, given an formula of circle like X^2+y^2=R^2

what's confusing me, who said that X is variable? maybe it's a set or something else? really every time I ready a formula "any formula" I stuck on this , who told us that X is a variable and not a set or not vector?!

 

[Dr.Peterson]

Hi guys, given an formula of circle like X^2+y^2=R^2

what's confusing me, who said that X is variable? maybe it's a set or something else? really every time I ready a formula "any formula" I stuck on this , who told us that X is a variable and not a set or not vector?!

"Who said?"

Whoever gives you the equation should tell you what the symbols refer to; namely, that x and y are the coordinates of a point on the plane lying on this circle, and R is the radius. If he doesn't do this, he hasn't really told you anything. You're absolutely right that they might be something other than numbers (though that's very unlikely).

You don't have to assume anything, and there doesn't have to be a universal rule that x is always a variable, or anything like that.

If someone states a formula without defining terms, you have every right (in fact, a need) to ask for the definitions -- just as we have a right and a need to ask you for particulars about your questions! This is all part of communication; a question without definitions is incomplete.

The other possibility is that the context in which this was stated would imply certain things. In a typical algebra book, in a chapter on graphing, x and y will always be coordinates of a point, and it doesn't have to be stated every time. When we say this is a "formula for a circle", that implies that x and y are coordinates of a point on the circle. So the second answer to your "who said" question is: the context. (This is, again, why we ask you to tell us the specific context of your questions.)

Look again at our guideline summary, and you will see these same points made: in order to communicate clearly, you need to give details and context. That is true not only when you ask us a question, but also when a teacher or author asks you to solve a problem! You need to look at everything that is said, and if not enough is said, ask the author.