Chapter 10

Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(\frac{7}{4}, \quad\) directrix \(y=6\)

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

1-4 Find the area of the region that is bounded by the given curve and lies in the specified sector. $$r=\theta^{2}, \quad 0 \leqslant \theta \leqslant \pi / 4$$

1-2 Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with \(r>0\) and one with \(r<0 .\) $$(a)(2, \pi / 3) \quad \text { (b) }(1,-3 \pi / 4) \quad \text { (c) }(-1, \pi / 2)$$

Find \(d y / d x\) $$x=t \sin t, \quad y=t^{2}+t$$

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. \(x=1+\sqrt{t}, \quad y=t^{2}-4 t, \quad 0 \leqslant t \leqslant 5\)

Write a polar equation of a conic with the focus at the origin and the given data. Parabola, directrix \(x=4\)

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$4 y+x^{2}=0$$

1-4 Find the area of the region that is bounded by the given curve and lies in the specified sector. $$r=e^{\theta / 2}, \quad \pi \leqslant \theta \leqslant 2 \pi$$

1-2 Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with \(r>0\) and one with \(r<0 .\) $$\begin{array}{lll}{\text { (a) }(1,7 \pi / 4)} & {\text { (b) }(-3, \pi / 6)} & {\text { (c) }(1,-1)}\end{array}$$

Find \(d y / d x\) $$x=1 / t, \quad y=\sqrt{t} e^{-t}$$

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. \(x=2 \cos t, \quad y=t-\cos t, \quad 0 \leq t \leqslant 2 \pi\)

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{3}{4}, \quad\) directrix \(x=-5\)

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

1-4 Find the area of the region that is bounded by the given curve and lies in the specified sector. $$r=\sin \theta, \quad \pi / 3 \leqslant \theta \leqslant 2 \pi / 3$$

\(3-4\) Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. $$ \begin{array}{lll}{\text { (a) }(1, \pi)} & {\text { (b) }(2,-2 \pi / 3)} & {\text { (c) }(-2,3 \pi / 4)}\end{array} $$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x=t^{4}+1, \quad y=t^{3}+t ; \quad t=-1$$

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. \(x=5 \sin t, \quad y=t^{2}, \quad-\pi \leqslant t \leqslant \pi\)

Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(2, \quad\) directrix \(y=-2\)

Find the vertex,focus, and directrix of the parabola and sketch its graph. $$y^{2}=12 x$$

1-4 Find the area of the region that is bounded by the given curve and lies in the specified sector. $$r=\sqrt{\sin \theta}, \quad 0 \leqslant \theta \leqslant \pi$$

\(3-4\) Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. $$(\text{a)}\quad(-\sqrt{2}, 5 \pi / 4) \quad(\text { b) } \quad(1,5 \pi / 2) \quad \text { (c) }(2,-7 \pi / 6)$$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x=t-t^{-1}, \quad y=1+t^{2} ; \quad t=1$$

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. \(x=e^{-t}+t, \quad y=e^{t}-t, \quad-2 \leqslant t \leqslant 2\)

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

\(5-6\) The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r>0\) and \(0 \leqslant \theta<2 \pi\) . (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r<0\) and \(0 \leqslant \theta<2 \pi\) $$(\text a)\quad(2,-2)$$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$\boldsymbol{x}=e^{\sqrt{t}}, \quad y=t-\ln t^{2} ; \quad t=1$$

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. \(x=3 t-5, \quad y=2 t+1\)

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity 0.8, vertex \((1, \pi / 2)\)

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$ x-1=(y+5)^{2} $$

\(5-6\) The Cartesian coordinates of a point are given. (i) Find polar coordinates \((r, \theta)\) of the point, where \(r>0\) and \(0 \leqslant \theta<2 \pi\) . (ii) Find polar coordinates \((r, \theta)\) of the point, where \(r<0\) and \(0 \leqslant \theta<2 \pi\) $$\begin{array}{ll}{\text { (a) }(3 \sqrt{3}, 3)} & {\text { (b) }(1,-2)}\end{array}$$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $$x=\cos \theta+\sin 2 \theta, \quad y=\sin \theta+\cos 2 \theta ; \quad \theta=0$$

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. \(x=1+t, \quad y=5-2 t, \quad-2 \leqslant t \leqslant 3\)

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{1}{2}, \quad\) directrix \(r=4 \sec \theta\)

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$y^{2}+2 y+12 x+25=0$$

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$1\leqslant r \leqslant 2$$

Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. $$x=1+\ln t, \quad y=t^{2}+2 ; \quad(1,3)$$

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. \(x=t^{2}-2, \quad y=5-2 t, \quad-3 \leqslant t \leqslant 4\)

Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(3, \quad\) directrix \(r=-6 \csc \theta\)

Find the vertex, focus, and directrix of the parabola and sketch its graph. $$y+12 x-2 x^{2}=16$$

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$r \geqslant 0, \quad \pi / 3 \leqslant \theta \leqslant 2 \pi / 3$$

Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. $$x=\tan \theta, \quad y=\sec \theta ; \quad(1, \sqrt{2})$$

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. \(x=1+3 t, \quad y=2-t^{2}\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{1}{1+\sin \theta}$$

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$0 \leqslant r<4, \quad-\pi / 2 \leqslant \theta<\pi / 6$$

\(9-14\) Sketch the curve and find the area that it encloses. $$ r=3 \cos \theta $$

Find an equation of the tangent(s) to the curve at the given point. Then graph the curve and the tangent(s). $$x=6 \sin t, \quad y=t^{2}+t ; \quad(0,0)$$

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as \(t\) increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. \(x=\sqrt{t}, \quad y=1-t\)

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{12}{3-10 \cos \theta}$$

\(9-14\) Sketch the curve and find the area that it encloses. $$ r=3(1+\cos \theta) $$