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- #1

- Feb 14, 2012

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$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0$

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3$

- Thread starter anemone
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- #1

- Feb 14, 2012

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$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0$

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3$

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- #2

- Jan 26, 2012

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By inspection, $a=0, b=-1$ solves the system. There may be other solutions.

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- #3

- Feb 14, 2012

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Thanks for participating, Ackbach!

Ahem... I can only give you at most half credit because your answer isn't complete!

My solution:

We're given

These two inequalities (*) and (**) suggest that $b<0$.

Now, if we let

$k=a+\sqrt{1+a^2}=f(a)>0$, $\rightarrow a=\dfrac{k^2-1}{2k}$

and substitute these two into the equation $\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0$, we get

$\dfrac{\dfrac{k^2-1}{2k}+b}{k}+b^2=0$

$\dfrac{\dfrac{k^2-1+2bk}{2k}}{k}+b^2=0$

$\dfrac{k^2-1+2bk}{2k^2}+b^2=0$

$k^2-1+2bk+2k^2b^2=0$

$(1+2b^2)k^2-(2b)k-1=0$

Solving it for $k$ by the quadratic formula, we obtain

$k=\dfrac{b\pm \sqrt{3b^2+1}}{1+2b^2}$

Since $k>0$, and $b<0$, we can say at this point that $k=\dfrac{b+ \sqrt{3b^2+1}}{1+2b^2}$, which in turn tells us we have only one value of $x$ as the answer.

Now, the only answer to this problem, like Ackbach has mentioned is $(x,y)=(0, -1)$.

Ahem... I can only give you at most half credit because your answer isn't complete!

My solution:

We're given

These two inequalities (*) and (**) suggest that $b<0$.

Now, if we let

$k=a+\sqrt{1+a^2}=f(a)>0$, $\rightarrow a=\dfrac{k^2-1}{2k}$

and substitute these two into the equation $\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0$, we get

$\dfrac{\dfrac{k^2-1}{2k}+b}{k}+b^2=0$

$\dfrac{\dfrac{k^2-1+2bk}{2k}}{k}+b^2=0$

$\dfrac{k^2-1+2bk}{2k^2}+b^2=0$

$k^2-1+2bk+2k^2b^2=0$

$(1+2b^2)k^2-(2b)k-1=0$

Solving it for $k$ by the quadratic formula, we obtain

$k=\dfrac{b\pm \sqrt{3b^2+1}}{1+2b^2}$

Since $k>0$, and $b<0$, we can say at this point that $k=\dfrac{b+ \sqrt{3b^2+1}}{1+2b^2}$, which in turn tells us we have only one value of $x$ as the answer.

Now, the only answer to this problem, like Ackbach has mentioned is $(x,y)=(0, -1)$.

Last edited:

- Jan 25, 2013

- 1,225

$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0---(1)$

$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0$

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3$

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3---(2)$

from (2) $b\neq 0$

from (1) it is easy to see a+b<0

from (2)$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3>\dfrac{a^2}{b^2}+2\sqrt{a^2}+b^2>3\times \left |a \right |\sqrt[3]{2}$(AP>GP)

$\therefore 1>\left | a \right |\sqrt[3]{2}$

if $a\in Z$ then a=0 and from (1)b=-1

Last edited:

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- Feb 14, 2012

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Hi$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0---(1)$

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3---(2)$

from (2) $b\neq 0$

from (1) it is easy to see a+b<0

from (2)$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3>\dfrac{a^2}{b^2}+2\sqrt{a^2}+b^2>3\times (\pm) a\sqrt[3]{2}$(AP>GP)

$\therefore 1>(\pm a)\sqrt[3]{2}$

if $a\in Z$ then a=0 and from (1)b=-1

Thanks for showing me another method to solve this problem and thanks too for participating. But I don't understand why there is a plus minus sign in front of the cube root of 2, after one applied the AM-GM to the terms $\dfrac{a^2}{b^2}, 2\sqrt{a^2}, b^2$:

$\dfrac{a^2}{b^2}+2\sqrt{a^2}+b^2>3(\pm \sqrt[3]{2})$

I see that even if $1>\pm\sqrt[3]{2}(a)$ is true, the solution set for $a$ that I can get from here is $-\dfrac{1}{\sqrt[3]{2}}<a<\dfrac{1}{\sqrt[3]{2}}$, but not what you have concluded... I'm sure I am missing something here, and I hope you will enlighten me.

- Jan 25, 2013

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we get a=0 (if a is an integer),and from (1) b=-1$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0---(1)$

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3---(2)$

from (2) $b\neq 0$

from (1) it is easy to see a+b<0

from (2)$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3>\dfrac{a^2}{b^2}+2\sqrt{a^2}+b^2>3\times \left |a \right |\sqrt[3]{2}$(AP>GP)

$\therefore 1>\left | a \right |\sqrt[3]{2}$

we get a=0 and from (1)b=-1

if a is not an integer ----(please wait )

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- #7

- Feb 7, 2012

- 2,785

Multiplying out the fractions, we can write the original equations as $$1+b + ab^2 + b^2c = 0, \qquad(1)$$ $$a^2 + 2b^2c + b^4 - 3b^2 = 0, \qquad (2)$$ where $c = \sqrt{1+a^2}$. Multiply (1) by $2$ and subtract (2) (to eliminate $c$): $$2a + 2b + 2ab^2 - a^2 - b^4 + 3b^2 = 0.$$ Write that as a quadratic in $a$ and solve it: $$a^2 - 2(1+b^2)a + (b^4 - 3b^2 - 2b) = 0,\qquad a = 1 + b^2 - \sqrt{5b^2 + 2b + 1} \qquad (3)$$ (we have to take the negative square root in (3) because $|a| < |b|$).

Using the definition of $c$, write (2) as $(c^2 - 1) + 2b^2c + b^4 - 3b^2 = 0$. Write that as a quadratic in $c$ and solve it: $$c^2 + 2cb^2 + (b^4 - 3b^2 - 1) = 0, \qquad c = -b^2 + \sqrt{3b^2+1} \qquad (4)$$ (this time, we have to take the positive square root because $c\geqslant 1$).

Now substitute (4) into (1) and solve for $a$: $$a+b + ab^2 + b^2\bigl(-b^2 + \sqrt{3b^2+1}\bigr), \qquad a = \frac{b^4 - b^2\sqrt{3b^2+1} - b}{1+b^2}. \qquad(5)$$

The next step is to equate the two expressions for $a$ ((3) and (5)), which gives an equation for $b$. After a small amount of simplification, this becomes: $$f(b) \overset{\text{def}}{=} 1+ b + 2b^2 + b^2\sqrt{3b^2+1} - (1+b^2) \sqrt{5b^2 + 2b + 1} = 0.$$

I don't see any hope of solving that equation, but a plot of the graph of $f(b)$ shows that its only zero in the interval $-1\leqslant b<0$ is when $b=-1$ (click on View attachment graph.pdf to see the printout).

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- #8

- Feb 14, 2012

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Thank youThis is a follow-up to anemone's solution above. She shows that $b<0$, and also that $a^2+b^4 \leqslant b^2$, from which $a^2 \leqslant b^2(1-b^2).$ It follows that $|a| < |b|$ and also that $|b| \leqslant 1.$ Therefore $-1\leqslant b<0.$

Multiplying out the fractions, we can write the original equations as $$1+b + ab^2 + b^2c = 0, \qquad(1)$$ $$a^2 + 2b^2c + b^4 - 3b^2 = 0, \qquad (2)$$ where $c = \sqrt{1+a^2}$. Multiply (1) by $2$ and subtract (2) (to eliminate $c$): $$2a + 2b + 2ab^2 - a^2 - b^4 + 3b^2 = 0.$$ Write that as a quadratic in $a$ and solve it: $$a^2 - 2(1+b^2)a + (b^4 - 3b^2 - 2b) = 0,\qquad a = 1 + b^2 - \sqrt{5b^2 + 2b + 1} \qquad (3)$$ (we have to take the negative square root in (3) because $|a| < |b|$).

Using the definition of $c$, write (2) as $(c^2 - 1) + 2b^2c + b^4 - 3b^2 = 0$. Write that as a quadratic in $c$ and solve it: $$c^2 + 2cb^2 + (b^4 - 3b^2 - 1) = 0, \qquad c = -b^2 + \sqrt{3b^2+1} \qquad (4)$$ (this time, we have to take the positive square root because $c\geqslant 1$).

Now substitute (4) into (1) and solve for $a$: $$a+b + ab^2 + b^2\bigl(-b^2 + \sqrt{3b^2+1}\bigr), \qquad a = \frac{b^4 - b^2\sqrt{3b^2+1} - b}{1+b^2}. \qquad(5)$$

The next step is to equate the two expressions for $a$ ((3) and (5)), which gives an equation for $b$. After a small amount of simplification, this becomes: $$f(b) \overset{\text{def}}{=} 1+ b + 2b^2 + b^2\sqrt{3b^2+1} - (1+b^2) \sqrt{5b^2 + 2b + 1} = 0.$$

I don't see any hope of solving that equation, but a plot of the graph of $f(b)$ shows that its only zero in the interval $-1\leqslant b<0$ is when $b=-1$ (click on View attachment 1690 to see the printout).

After reading your post, I realized I have provided a false solution. My apologies.

I will stand in the corner with my nose against the wall for making the silly assumption as follows:

In $k=\dfrac{b\pm \sqrt{3b^2+1}}{1+2b^2}$, since $k>0$, and $b<0$, we can say at this point that $k=\dfrac{b+ \sqrt{3b^2+1}}{1+2b^2}$, which in turn tells us we have only one value of $x$ as the answer".

- Jan 25, 2013

- 1,225

from (1) a+b<0Solve the simultaneous equation and find the values of $a$ and $b$.

$\dfrac{a+b}{a+\sqrt{1+a^2}}+b^2=0$---(1)

$\dfrac{a^2}{b^2}+2\sqrt{1+a^2}+b^2=3---(2)$

from (2) $a^2<b^2,\therefore b<a,\,\, (a-b)>0$$\therefore b<0$

the range of a : -1<a<1---(3)

the range of b:$-1\leq b<0$---(4)

let $\sqrt {1+a^2}=1+x(x\geq 0)$

we get $a^2=x^2+2x$

from (2) $\dfrac {x^2+2x}{b^2}+2+2x+b^2=3$

$\dfrac {x^2+2x+2b^2x+b^4}{b^2}=1=\dfrac{b^2}{b^2}$

$(x^2+2x)+b^2(b^2+2x)=b^2$

compare both side we have:

$(x^2+2x)=0 ,\,\, and (b^2+2x)=1$

x=0 or x=-2 (deleted)

so we get a=0 and b=-1 or (b=1 also deleted)

$\therefore$ (a,b)=(0,-1) is the only solution

Last edited:

- Mar 31, 2013

- 1,346

$(x^2+2x)+b^2(b^2+2x)=b^2$from (1) a+b<0

from (2) $a^2<b^2,\therefore b<a,\,\, (a-b)>0$$\therefore b<0$

the range of a : -1<a<1---(3)

the range of b:$-1\leq b<0$---(4)

let $\sqrt {1+a^2}=1+x(x\geq 0)$

we get $a^2=x^2+2x$

from (2) $\dfrac {x^2+2x}{b^2}+2+2x+b^2=3$

$\dfrac {x^2+2x+2b^2x+b^4}{b^2}=1=\dfrac{b^2}{b^2}$

$(x^2+2x)+b^2(b^2+2x)=b^2$

compare both side we have:

$(x^2+2x)=0 ,\,\, and (b^2+2x)=1$

x=0 or x=-2 (deleted)

so we get a=0 and b=-1 or (b=1 also deleted)

$\therefore$ (a,b)=(0,-1) is the only solution

compare both side we have:

$(x^2+2x)=0 ,\,\, and (b^2+2x)=1$

this statement is not correct. This is correct if it is true for any b and not a particular b.

- Jan 25, 2013

- 1,225

please note the range of a,and b$(x^2+2x)+b^2(b^2+2x)=b^2$

compare both side we have:

$(x^2+2x)=0 ,\,\, and (b^2+2x)=1$

this statement is not correct. This is correct if it is true for any b and not a particular b.

in fact b=$\pm 1,\,\, and\,\, b=\pm \sqrt 5$

but b=1, and b=$\pm \sqrt 5$ must be deleted

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- #12

- Feb 7, 2012

- 2,785

Very nice solution. I would prefer to simplify the final part of it by writing $x^2+2x+2b^2x+b^4 = b^2$ as $(x+b^2)^2 + 2x = b^2$. Since $x\geqslant0$, the left side is clearly greater than the right unless $x=0$.from (1) a+b<0

from (2) $a^2<b^2,\therefore b<a,\,\, (a-b)>0$$\therefore b<0$

the range of a : -1<a<1---(3)

the range of b:$-1\leq b<0$---(4)

let $\sqrt {1+a^2}=1+x(x\geq 0)$

we get $a^2=x^2+2x$

from (2) $\dfrac {x^2+2x}{b^2}+2+2x+b^2=3$

$\dfrac {x^2+2x+2b^2x+b^4}{b^2}=1=\dfrac{b^2}{b^2}$

$(x^2+2x)+b^2(b^2+2x)=b^2$

compare both side we have:

$(x^2+2x)=0 ,\,\, and (b^2+2x)=1$

x=0 or x=-2 (deleted)

so we get a=0 and b=-1 or (b=1 also deleted)

$\therefore$ (a,b)=(0,-1) is the only solution

That proof relies entirely on information from the second of the two given equations; the first one is not needed at all!

- Mar 31, 2013

- 1,346

some how I am missing the clearly greater than the right part.Very nice solution. I would prefer to simplify the final part of it by writing $x^2+2x+2b^2x+b^4 = b^2$ as $(x+b^2)^2 + 2x = b^2$. Since $x\geqslant0$, the left side is clearly greater than the right unless $x=0$.

That proof relies entirely on information from the second of the two given equations; the first one is not needed at all!

if b^2 < 1 then b^4 < b^2 in that case x could be positive. I can neither prove it nor disprove it.

any help in this regards

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- #14

- Feb 7, 2012

- 2,785

You are quite right – I was confusing $b^2$ with $b^4$ at some point, and my "simplification" does not work.some how I am missing the clearly greater than the right part.

if b^2 < 1 then b^4 < b^2 in that case x could be positive. I can neither prove it nor disprove it.

any help in this regards