Question

va rog din suflet.....

Answer 1

$\it (ay-bx)^2\geq0 \Leftrightarrow a^2y^2+b^2x^2-2abxy\geq0 \Leftrightarrow \\ \\\Leftrightarrow a^2y^2+b^2x^2 \geq2abxy|_{+(abx^2+aby^2)} \Leftrightarrow abx^2+aby^2+a^2y^2+b^2x^2\geq \\ \\ \geq abx^2+aby^2+ 2abxy \Leftrightarrow (abx^2+a^2y^2)+(b^2x^2+aby^2)\geq ab(x+y)^2\Leftrightarrow \\ \\\Leftrightarrow a(bx^2+ay^2)+b(bx^2+ay^2)\geqab(x+y)^2\Leftrightarrow(bx^2+ay^2)(a+b)\geqab(x+y)^2\Leftrightarrow$

$\it \Leftrightarrow\dfrac{bx^2+ay^2}{ab} \geq\dfrac{(x+y)^2}{a+b} \Leftrightarrow\dfrac{x^2}{a}+\dfrac{y^2}{b} \geq\dfrac{(x+y)^2}{a+b}$