Find and equation equivalent to x^2-y^2=4 in polar coordinates

Answer:[tex]\text{The equation is }r=\frac{2}{\sqrt{(1-2\sin^2\theta)}}[/tex]Step-by-step explanation:Given the equation[tex]x^2-y^2=4[/tex]we have to find the equation in polar coordinates.To convert in polar coordinates, we have to put [tex]x=r\cos\theta[/tex][tex]y=r\sin\theta[/tex][tex]\text{in given equation }x^2-y^2=4[/tex][tex](r\cos\theta)^2-(r\sin\theta)^2=4[/tex][tex]r^2\cos^2\theta-r^2\sin^2\theta=4[/tex][tex]r^2(\cos^2\theta-\sin^2\theta)=4[/tex][tex]As, \sin^2\theta+\cos^2\theta=1[/tex][tex]gives\thinspace \cos^2\theta=1-\sin^2\theta[/tex][tex]r^2(1-\sin^2\theta)-r^2\sin^2\theta=4[/tex][tex]r^2-r^2\sin^2\theta-r^2\sin^2\theta=4[/tex][tex]r^2-2r^2\sin^2\theta=4[/tex][tex]r^2(1-2\sin^2\theta)=4[/tex][tex]r^2=\frac{4}{(1-2\sin^2\theta)}[/tex]Take square root on both sides [tex]r=\frac{\sqrt4}{\sqrt{(1-2\sin^2\theta)}}[/tex][tex]r=\frac{2}{\sqrt{(1-2\sin^2\theta)}}[/tex]which is required equation