91Ó°ÊÓ

Sketch the ellipse, and label the foci, vertices, and ends of the minor axis. (a) \((x+3)^{2}+4(y-5)^{2}=16\) (b) \(\frac{1}{4} x^{2}+\frac{1}{9}(y+2)^{2}-1=0\)

Find polar coordinates of all points at which the polar curve has a horizontal or a vertical tangent line. $$r=a(1+\cos \theta)$$

Identify the curve by transforming the given polar equation to rectangular coordinates. (a) \(r=2\) (b) \(r \sin \theta=4\) (c) \(r=3 \cos \theta\) (d) \(r=\frac{6}{3 \cos \theta+2 \sin \theta}\)

Rotate the coordinate axes to remove the \(x y\) -term. Then identify the type of conic and sketch its graph. $$x^{2}-24 x y+16 y^{2}-80 x-60 y+100=0$$

Sketch the curve by eliminating the parameter, and indicate the direction of increasing \(t\) $$x=\cos 2 t, y=\sin t \quad(-\pi / 2 \leq t \leq \pi / 2)$$

Identify the curve by transforming the given polar equation to rectangular coordinates. (a) \(r=5 \sec \theta\) (b) \(r=2 \sin \theta\) (c) \(r=4 \cos \theta+4 \sin \theta\) (d) \(r=\sec \theta \tan \theta\)

Find polar coordinates of all points at which the polar curve has a horizontal or a vertical tangent line. $$r=a \sin \theta$$

Rotate the coordinate axes to remove the \(x y\) -term. Then identify the type of conic and sketch its graph. $$5 x^{2}-6 x y+5 y^{2}-8 \sqrt{2} x+8 \sqrt{2} y=8$$

Sketch the curve by eliminating the parameter, and indicate the direction of increasing \(t\) $$x=4 t+3, y=16 t^{2}-9$$

Sketch the ellipse, and label the foci, vertices, and ends of the minor axis. (a) \(9 x^{2}+4 y^{2}-18 x+24 y+9=0\) (b) \(5 x^{2}+9 y^{2}+20 x-54 y=-56\)