Ellipse region in polar coordinates

Question

if I want to write the region in $R^2$ bounded by the ellipse

$$10x^2 + 17 y^2 = 29$$

In polar coordinates($x=r\cos \theta, y= r \sin \theta$), how can I find the limit of $r$?

Answer 1

Substituting the polar coordinates you get $$ r^2(10\cos^2\theta+17\sin^2\theta)=29\implies r=\frac{\sqrt{29}}{\sqrt{10\cos^2\theta+17\sin^2\theta}}. $$ In this type of problem, it is ussually easier to work with modified polar coordinates: $$ x=a\,r\cos\theta,\quad y=b \,r\sin\theta. $$ In this examople $$ a=\sqrt{\frac{29}{10}},\quad b=\sqrt{\frac{29}{17}}. $$