Prove that $(a^2 + b^2)(x^2 + y^2) \geq (ax + by)^2$, with equality if and only if $a: b = x: y$.
I have tried proof by induction but am stuck as I am confused on how to use ratios. I have also expanded the brackets:
$$a^2x^2 + a^2y^2 + b^2x^2 + b^2y^2 \geq a^2x^2 + 2abxy + b^2y^2.$$
Now, I have simplified my expansion to $(ay-bx)(ay-bx) \geq 0$
I have also realised that, for $(ay-bx)(ay-bx) \geq 0$, $(ay-bx)$ must be either greater than, less than or equal to $0$.
In the first case, $ay>bx$. In the second case, $ay<bx$ and in the third case, $ay=bx$
Yet again, the ratios stump me.
How can I proceed?
Thank you.