Find the minimum and maximum distances from point $(2,6)$ to ellipse $9x^2+8y^2-36x-16y-28=0$

Question

Having trouble with this. I'm not getting any ideas.

Find the minimum and maximum distances of point $(2,6)$ from the ellipse $$9x^2+8y^2-36x-16y-28=0$$

Answer 1

hint

Your ellipse has the equation

$$9(x-2)^2+8(y-1)^2=72$$ or

$$(\frac {x-2}{\sqrt {8}})^2+(\frac {y-1}{3})^2=1$$

it can be paremetrized by

$$x=\sqrt {8}\cos (t)+2$$ $$y=3\sin (t)+1$$

the square of the distance from the point $(2,6) $ to a point of the ellipse is

$$D^2(t)=8\cos^2 (t)+9\sin^2 (t)-30\sin (t)+25$$ $$=\sin^2 (t)-30\sin (t)+33$$ $$=(\sin (t)-1)(\sin (t)+1-30)+4\ge 4$$

Your distance is $D(\frac {\pi}{2})=2$ .