MHB Find the condition for equality to hold

[anemone]

Hi,
I've encountered a problem in deciding the condition in order for the equality to hold.
Here is the problem:

If $x\sqrt {1-y^2} + y \sqrt {1-x^2}=1$, prove that $x^2+y^2=1$

By using the Cauchy-Schwarz inequality, it's fairly easy to prove that $x\sqrt {1-y^2} + y \sqrt {1-x^2}\leq1$

Next, what I tried to do is to work backwards and let $x^2+y^2=1$, then I see that $x=\sqrt {1-y^2}$. After making that substitution into the LHS of the inequality $ x\sqrt {1-y^2} + y \sqrt {1-x^2} $ and I eventually get 1 as the final answer.

What do you think, Sir? I feel bad for doing this.

Do you have any idea to deduce the condition from $x\sqrt {1-y^2} + y \sqrt {1-x^2}\leq1$?

Thanks.

 

[alexmahone]

Hi,
I've encountered a problem in deciding the condition in order for the equality to hold.
Here is the problem:

If $x\sqrt {1-y^2} + y \sqrt {1-x^2}=1$, prove that $x^2+y^2=1$

By using the Cauchy-Schwarz inequality, it's fairly easy to prove that $x\sqrt {1-y^2} + y \sqrt {1-x^2}\leq1$

In the Cauchy Schwarz inequality, equality holds only if

$\displaystyle \frac{x}{\sqrt{1-x^2}}=\frac{\sqrt{1-y^2}}{y}$

$\displaystyle xy=\sqrt{(1-x^2)(1-y^2)}$

$\displaystyle x^2y^2=(1-x^2)(1-y^2)=1-x^2-y^2+x^2y^2$

$\displaystyle x^2+y^2=1$

 

[anemone]

Gosh, I missed that part of definition!

Thanks, Alexmahone.