91Ó°ÊÓ

Find the equation of the hyperbola that satisfies the given conditions. Center (0,0)\(;\) vertex (2,0)\(;\) passing through \((4, \sqrt{3})\)

Convert the rectangular coordinates to polar coordinates. $$(1,1)$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$3 x^{2}+2 \sqrt{3} x y+y^{2}+4 x-4 \sqrt{3} y-16=0$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(x\) -axis; major axis of length \(12 ;\) minor axis of length \(8 .\)

Sketch the graph of the equation and label the vertices. $$r=\frac{5}{1+\cos \theta}$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=8 t^{3}-4 t^{2}+3, \quad y=2 t-4, \quad \text { any real number } t$$

Convert the rectangular coordinates to polar coordinates. $$(\sqrt{2},-\sqrt{2})$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$3 x^{2}+2 \sqrt{2} x y+2 y^{2}-12=0$$

Find the equation of the hyperbola that satisfies the given conditions. Center (0,0)\(;\) vertex \((0, \sqrt{12}) ;\) passing through \((2 \sqrt{3}, 6)\)

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(y\) -axis; major axis of length \(20 ;\) minor axis of length 18.