91Ó°ÊÓ

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(1,1), \quad \phi=45^{\circ}$$

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=2 t, \quad y=t+6$$

\(1-8=\) Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{2}{3},\) directrix \(x=3\)

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(-2,1), \quad \phi=30^{\circ}$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=6 t-4, \quad y=3 t, \quad t \geq 0$$

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{(x-3)^{2}}{16}+(y+3)^{2}=1$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(\frac{4}{3},\) directrix \(x=-3\)

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(3,-\sqrt{3}), \quad \phi=60^{\circ}$$

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=t^{2}, \quad y=t-2, \quad 2 \leq t \leq 4$$