# Simultaneous equations

1. Jan 12, 2010

### Mentallic

I need to solve these 2 equations:

$$r^2-a^2=\sqrt{(r^2-a^2)(R^2-(b-d)^2)}$$

$$\sqrt{(r^2-a^2)(R^2-(d-b)^2)}=R^2-d^2+bd-ab+ad$$

Now wait, before you run off thinking "screw that - too many letters, too much hassle" I don't actually want you to solve this for me, but I have a question about it.

Firstly, r, R and d are constants and I'm looking to solve for a and b so what I essentially want is:

$$a=f(r,R,d)$$ and $$b=f(r,R,d)$$ (I'm not even sure if I have expressed what I want properly, I just took a best guess at this. It translates into "a as a function in terms of r, R and d").

What I thought I'd try is, instead of expanding one equation and solving a quadratic in terms of one of the variables a or b, I'll eliminate the $$\sqrt{(r^2-a^2)(R^2-(d-b)^2)}$$ so what I end up having is:

$$R^2-d^2+bd-ab+ad=r^2-a^2$$

but then this doesn't look like it's helping me solve for either variable. Is the only way I can solve for a and b by going through the long painful process of using the quadratic formula? Any other suggestions on a simpler approach I could take?

2. Jan 13, 2010

### Staff: Mentor

Set r2 - a2 = R2 - d2 + bd - ab + ad

The radicals in the two equations are equal; the only difference is that one has a factor of (b - d)2 inside, while the other has a factor of (d - b)2 inside, and these two factors are equal.

3. Jan 13, 2010

### Mentallic

Yes I realized that and I already had done what you suggested. It's near the end of my OP.

I guess I can solve for either a or b in this 3rd equation and then substitute that back into the 1st or 2nd equation. It doesn't look impossible to solve, but I know it's going to be a pain...

4. Jan 13, 2010

### Mentallic

Yep, as I thought - my only hope was that many crucial terms would cancel nicely, but there wasn't enough of that happening to help me solve these equations. I have resulted in a 5th order equation in the variable a, so it's quite fruitless to keep going.

It seems like all my most recent math problems have been cut short of being solved simply by becoming too overwhelming to work with. Maybe I should quit with giving myself problems to solve.

5. Jan 13, 2010

### bikengr

6. Jan 13, 2010

### Mentallic

I beg to differ.

How so?

You'd be the 3rd person in this thread that has realized this and posted it :tongue:

Only a will be negative (although it doesn't have to be the case for the question), the only real restrictions are $r,R>0$

This would be the case if you hadn't incorrectly made the equality $$\sqrt{(r^2-a^2)(R^2-(b-d)^2)}=R^2-(b-d)^2$$ I think?

$a=b$ or $d=b$ iff $r,R,b=0$ and this is a pretty boring case and makes solving the problem quite futile.

EDIT: I now see what you did in the first equation, I'm going to check over everything you've done again to see where else the problem may lie.

7. Jan 13, 2010

### Mentallic

Haha it seems everything you've done was correct.

But my statement still holds that $a\neq b$ and $b\neq d$ so that can only mean my equations are wrong.

oops

Thanks bikengr!

8. Jan 13, 2010

### D H

Staff Emeritus
$$r^2-a^2=\sqrt{(r^2-a^2)(R^2-(b-d)^2)}$$

Square both sides:

$$\left(r^2-a^2\right)^2=(r^2-a^2)(R^2-(b-d)^2)$$

If $$r^2-a^2 \ne 0$$ you can clear that common factor, leaving

$$r^2-a^2=(R^2-(b-d)^2$$

and, as bikengr showed, this eventually leads to $$(a-b)(d-b)=0$$.

However, that ignores the other solution that results from [tex]r^2-a^2=0[tex].

9. Jan 13, 2010

### bikengr

In my frustration at login difficulties, and also general sleep deprivation, I didn't state the solution that I found all that clearly:

The quantities must be chosen to satisfy r^2-a^2 = R^2-(b-d)^2
Furthermore, either a=b or d=b
[I am not precluding other solutions due to positive/negative square roots, and also restrictions to allow the square roots to be real. I simply haven't looked into them.]

These two relations in 5 quantities will generally allow you to choose three variables relatively freely, and solve for two others. For example select d, b and r arbitrarily. The second relation tells you the value of a. The first relation lets you solve for R. Note that there are limits to the choices of d,b,r if you want R to be real. R will not be real if (b-d)^2 and r^2 are small, and a^2 is large.

10. Jan 13, 2010

### D H

Staff Emeritus
Not necessarily. In finding that you implicitly divided by (r2 - a2). This division is not valid if a2=r2. Setting a2=r2 leads to a new family of solutions.

11. Jan 13, 2010

### bikengr

Just so. I tried to be quite explicit that I probably did not have all solutions. My main aim was to break the logjam, in which proposer was tied in knots about high order polynomials. I think I have done so, however cavalierly, and probably won't be adding more.

12. Jan 20, 2010

### zgozvrm

I followed you, up to this point. But how do you cancel R^2? We don't know the relationship between r and R, do we?