Question

Is the point $(4,-3)$ within or without the ellipse $5x^2+7y^2={11}^2$?

Answer 1

Let $f(x,y)=5x^2+7y^2-121$

$f(4,-3)=\mathrm{5.4.4}+\mathrm{7.3.3}-121=28$

This is positive so the point lies outside the ellipse.