4x4 equation

[UwuUwuwuwu]

Are there any neat/simple ways to solve the below 4x4 equation?.. I have been struggling with for like 3 hours..
Thank you... (Here is a little demonstration of my nonsense work) Edit: I only need to find the ratio (yA-yB) /(xA-xB)

 

[Dr.Peterson]

The pictures are making my head spin!

But the first thing I'd do is get rid of the subscripts to make the writing easier. I'll let [MATH]w=x_A, x=x_B, y=y_A, z=y_B[/MATH].

Then the equations are (if I've copied correctly)

[MATH](w-3)^2 + (y-2)^2 = 25[/MATH]​

[MATH](x-3)^2 + (z-2)^2 = 25[/MATH]​

[MATH](w-x)^2 + (y-z)^2 = 36[/MATH]​

[MATH](w+x-6)^2 + (y+z-4)^2 = 64[/MATH]​

I haven't tried solving, but one thought I have is to make a change of variables (not just names) to hide the 3 and 2:

[MATH]a = w-3[/MATH]​

[MATH]b=x-3[/MATH]​

[MATH]c=y-2[/MATH]​

[MATH]d=z-2[/MATH]​

Give that a try, and see if I'm right about how much it will help!

 

[UwuUwuwuwu]

So far, I only got ab+cd=7. Edit: Original exercise : Let a circle with a center K(3,2) and r=5.Find the line in which the midpoints of the chords of the circle with a length of 4 move. My deep apologies for this interpration.. Hope you understand Something.. P.S I got confused with this exercise because it does not make clear if the chords have a length of 4 or the line. The numbers are not the same in each case..

 

[Otis]

… I only need to find the ratio (yA-yB) /(xA-xB)

Hello UwuUwuwuwu. There are many values for that ratio because the system has a lot of solutions. (I checked, using computer software.)

Some of those solutions are Rational numbers, and some are Irrational. There are also Complex solutions containing imaginary numbers.

I'm thinking that you're not interested in all possible values for (yA-yB)/(xA-xB). Do you have any additional information?

Here's just one of the many different Irrational results:

xA = 3 - 2 · √6
yA = 1
xB = 99/25 - (14/25)(√6)
yB = 43/25 - (48/25)(√6)

?

 

[Dr.Peterson]

Original exercise : Let a circle with a center K(3,2) and r=5.Find the line in which the midpoints of the chords of the circle with a length of 4 move. My deep apologies for this interpration.. Hope you understand Something..

P.S I got confused with this exercise because it does not make clear if the chords have a length of 4 or the line. The numbers are not the same in each case..

Is that the exact wording of the original problem? It doesn't quite make sense; if I take it literally, then the midpoint of any chord with length 4 will like on a concentric circle that is easy to find, not on a line.

In your equations, you seem to have made the chord have length 6, not 4, and assumed the midpoint of the chord lies on a circle with radius 4, which would be correct for these numbers -- but that would be deducible from the other facts, making it a dependent system -- which explains why the equations have many solutions!

And your equations don't answer any particular question, since you already know the locus of the midpoint. I think we need to see the entire problem exactly as given to you. But if "line" means "curve" or "locus", then it may be that the answer is the circle that led to your last equation, (x-3)^2 + (y-2)^2 = 16.

 

[UwuUwuwuwu]

So :

 

[Dr.Peterson]

This appears to be two different interpretations of the problem, not the exact wording of the problem itself, which I asked for. That doesn't help! (It also doesn't help that I can't quite read it all.)

Please show an image of the problem itself, even if it is in another language and you have to translate it in addition.

 

[UwuUwuwuwu]

23 a...It does not help Ik.. The thing is that I cannot understand 100% either. Translation : (word by word). Let a circle with a center K(3,2) and a radius r=5.Find the line in which the midpoints of the chords of the circle move with a length of 4.

 

[JeffM]

The picture is too small to be legible, but I don't know Greek so that makes no difference.

Let's see if we can clarify the problem.

Is the problem about chords of length 4 or about chords that are distant from the center of the circle by a length of 4?

 

[UwuUwuwuwu]

The picture is too small to be legible, but I don't know Greek so that makes no difference.

Let's see if we can clarify the problem.

Is the problem about chords of length 4 or about chords that are distant from the center of the circle by a length of 4?

I cannot understand it either.. Lol..

 

[Dr.Peterson]

I think you just have to ask your teacher or whoever wrote this! Trying to read the Greek with my little knowledge, and with the help of Google, I can only agree with you; it seems most likely to mean that the chord length is 4, so that the locus is the circle with radius [MATH]\sqrt{21}[/MATH].

 

[UwuUwuwuwu]

I think you just have to ask your teacher or whoever wrote this! Trying to read the Greek with my little knowledge, and with the help of Google, I can only agree with you; it seems most likely to mean that the chord length is 4, so that the locus is the circle with radius [MATH]\sqrt{21}[/MATH].

Right... But numbers do not make such a huge difference to the problem.

 

[UwuUwuwuwu]

Update:

Is that the exact wording of the original problem? It doesn't quite make sense; if I take it literally, then the midpoint of any chord with length 4 will like on a concentric circle that is easy to find, not on a line.

In your equations, you seem to have made the chord have length 6, not 4, and assumed the midpoint of the chord lies on a circle with radius 4, which would be correct for these numbers -- but that would be deducible from the other facts, making it a dependent system -- which explains why the equations have many solutions!

And your equations don't answer any particular question, since you already know the locus of the midpoint. I think we need to see the entire problem exactly as given to you. But if "line" means "curve" or "locus", then it may be that the answer is the circle that led to your last equation, (x-3)^2 + (y-2)^2 = 16.

I think you just have to ask your teacher or whoever wrote this! Trying to read the Greek with my little knowledge, and with the help of Google, I can only agree with you; it seems most likely to mean that the chord length is 4, so that the locus is the circle with radius [MATH]\sqrt{21}[/MATH].

Yep he told me that '' means curve''. And in this case the line is the circle with center K(3,2) and r=sqrt(21).... Anyways, everyone' s help has been really useful and made me understand the problem better.