Points on the curve

Question

We have to find points on the curve $ax^2+ay^2+2 bxy=c$ (Where c>b>a ) whose distance from origin is minimum .

I am not getting any start . I am able to just find that the curve would be hyperbola

Answer 1

Draw a circle radius $r$ centre the origin. Its equation is $x^2+y^2=r^2\ (1)$. If it intersects the hyperbola at $(x,y)$ then we have $ar^2+2bxy=c$, so $x^2y^2=\left(\frac{c-ar^2}{2b}\right)^2\ (2)$. Since $x^2,y^2$ are positive numbers we can apply the arithmetic/geometric mean result to (1) and (2) to get $\left(\frac{r^2}{2}\right)^2\ge\left(\frac{c-ar^2}{2b}\right)^2$ and hence $r^2\ge\frac{c}{a+b}$.

So the minimum distance is $\sqrt{\frac{c}{a+b}}$ and satisfied by points $(x,y)$ such that $x^2=y^2=\frac{c}{2(a+b)}$. Checking with the equation of the curve, we see that the signs of $x,y$ must be the same, so the two points are $(\sqrt{\frac{c}{2(a+b)}},\sqrt{\frac{c}{2(a+b)}})$ and $(-\sqrt{\frac{c}{2(a+b)}},-\sqrt{\frac{c}{2(a+b)}})$.