Chapter 10

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. $$x=2 t, \quad y=t+6$$

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. $$\frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

\(1-8=\) 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 \(x=3\)

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. $$x=6 t-4, \quad y=3 t, \quad t \geq 0$$

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

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 \(x=-3\)

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. $$x=t^{2}, \quad y=t-2, \quad 2 \leq t \leq 4$$

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

Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$\frac{x^{2}}{9}+\frac{(y+5)^{2}}{25}=1$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, directrix \(y=2\)

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. $$x=2 t+1, \quad y=\left(t+\frac{1}{2}\right)^{2}$$

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(2,0), \quad \phi=15^{\circ}$$

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

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{1}{2}, \operatorname{directrix} y=-4\)

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. $$x=\sqrt{t}, \quad y=1-t$$

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(0,2), \quad \phi=55^{\circ}$$

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$(x-3)^{2}=8(y+1)$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(4,\) directrix \(r=5 \sec \theta\)

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{x^{2}}{4}-\frac{y^{2}}{16}=1$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$

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. $$x=t^{2}, \quad y=t^{4}+1$$

Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ}$$

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$(y+5)^{2}=-6 x+12$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6,\) directrix \(r=2 \csc \theta\)

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$$

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. $$x=\frac{1}{t}, \quad y=t+1$$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}-3 y^{2}=4, \quad \phi=60^{\circ}$$

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$-4\left(x+\frac{1}{2}\right)^{2}=y$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$y^{2}-\frac{x^{2}}{25}=1$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$9 x^{2}+4 y^{2}=36$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$y^{2}=4 x$$

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. $$x=t+1, \quad y=\frac{t}{t+1}$$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$y=(x-1)^{2}, \quad \phi=45^{\circ}$$

Find the vertex, focus, and directrix of the parabola, and sketch the graph. $$y^{2}=16 x-8$$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.4,\) vertex at \((2,0)\)

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$x^{2}-y^{2}+4=0$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$4 x^{2}+25 y^{2}=100$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}=y$$

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. $$x=4 t^{2}, \quad y=8 t^{3}$$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}-y^{2}=2 y, \quad \phi=\cos ^{-1} \frac{3}{5}$$

Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$\frac{(x+1)^{2}}{9}-\frac{(y-3)^{2}}{16}=1$$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$x^{2}-y^{2}=1$$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$x^{2}+4 y^{2}=16$$

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x^{2}=9 y$$

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. $$x=|t|, \quad y=|1-| t||$$

Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$x^{2}+2 y^{2}=16, \quad \phi=\sin ^{-1} \frac{3}{5}$$

Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$(x-8)^{2}-(y+6)^{2}=1$$