91Ó°ÊÓ

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$17 x^{2}-12 x y+8 y^{2}-80=0$$

Find the equation of the hyperbola that satisfies the given conditions. Center (0,0)\(; x\) -intercepts \(\pm 3 ;\) asymptote \(y=2 x\)

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$3 x-2=(y+3)^{2}$$

In Exercises \(17-28,\) determine the vertex, focus, and directrix of the parabola without graphing and state whether it opens upward, downward, left, or right. $$2 x-1=-6(y+1)^{2}$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t^{2}+1, \quad y=t^{2}-1, \quad \text { any real number } t$$

Convert the rectangular coordinates to polar coordinates. $$(2 \sqrt{3}, 2)$$

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$11 x^{2}-24 x y+4 y^{2}+30 x+40 y-45=0$$

Find the equation of the ellipse that satisfies the given conditions. Center (0,0)\(;\) foci on \(y\) -axis; \(x\) -intercepts \(\pm 1 ; y\) -intercepts \(\pm 8.\)

Sketch the graph of the equation and label the vertices. $$r=\frac{5}{3+2 \sin \theta}$$

The given curve is part of the graph of an equation in \(x\) and \(y .\) Find the equation by eliminating the parameter. $$x=t^{3}-3 t^{2}+2 t, \quad y=t-1, \quad \text { any real number } t$$