SOLUTION: pls.. help me in this problem. Use the properties of proportionality to solve: {{{a+x+sqrt(a^2-x^2)/a+x-sqrt(a^2-x^2)=b/x

Question 550989: pls.. help me in this problem.
Use the properties of proportionality to solve:
a+x+sqrt(a^2-x^2)/a+x-sqrt(a^2-x^2)=b/x
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I assume your equation is

Multiplying both sides of the equal sign by the product of the denominators we get an (almost) equivalent equation stating that the cross products are equal. I assume that's what was meant by "Use the properties of proportionality."

(Multiplying by an expression that includes variables risks introducing extraneous solutions, solutions of the new equations that are not solutions of the original equation).

Now we square both sides of the equation. (By doing that we may be introducing extraneous solutions too).

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The expression above implies , ensuring that the square roots in the original equation will have a real value.
It also results in the expression , which includes all the solutions of the original equation, plus extraneous solutions.
If those tentative solutions were constant numbers, or simpler expressions, we could substitute and verify. In this case, it's easier to hunt down and eliminate the extraneous solutions the hard way.
Extraneous solutions include those that make the denominators zero.
There are no solutions for or , which make zero.
For , the expression found for turns into
, and , but makes a denominator in the original equation zero, so only is a solution.
For to have a real, positive value, it must be , or .
However, there are solutions only if , or .
For gives extraneous solutions, that make just one of the factors in negative.
For , it gives extraneous solutions that make negative exactly three of the factors in .
Those extraneous solutions are solutions of that were introduced on squaring both sides.
THE SOLUTION
and
for , or
(examples: (a,b)=(5,8), (a,b)=(-5,-2)) }.
For other (a,b) pairs there are no solutions (examples: (a,b)=(-5,-8), (a,b)=(-5,-2) ).
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