＜Senior High School Math＞: inequality question

$(1).omit$
$(2).(a^2-b^2)(\frac{x^2}{a^2}-\frac{y^2}{b^2})=(x^2+y^2)-(\frac{a^2{y}^2}{b^2}+\frac{b^2{x}^2}{a^2})\le x^2+y^2-2xy=(x-y{)}^2$

• when $\frac{ay}{b}=\frac{bx}{a}$, the equation is satisfied.

$(3).$ $set:\sqrt{3m-5}=x,\sqrt{m-2}=y$
$then:x^2-3y^2=1$
$have:\frac{x^2}{1}-\frac{y^2}{\frac{1}{3}}=1,$ according to the question(2), we get:
$(x-y{)}^2\ge (1-\frac{1}{3})(\frac{x^2}{1}-\frac{y^2}{\frac{1}{3}})=\frac{2}{3}$

- Pay attention to equal conditions!!!

• we can click here to see some more examples about the Cauchy-inequality.