Chapter 11

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ (x-1)^{2}+y^{2}=1 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 5)\) conjugate axis of length 4

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex \(V(-1,0)\) focus \(F(-4,0)\)

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{3}{4}\), vertices \(V(0, \pm 4)\)

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=1, \quad r \cos \theta=5 $$

(a) Describe the graph of a curve \(C\) that has the parametrization $$ x=-2+3 \sin t, \quad y=3-3 \cos t, \quad 0 \leq t \leq 2 \pi . $$ (b) Change the parametrization to $$ x=-2-3 \sin t, \quad y=3+3 \cos t ; \quad 0 \leq t \leq 2 \pi $$ and describe how this changes the graph from part (a). (c) Change the parametrization to $$ x=-2+3 \sin t, \quad y=3+3 \cos t ; \quad 0 \leq t \leq 2 \pi $$ and describe how this changes the graph from part (a).

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ (x+2)^{2}+(y-3)^{2}=13 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V(\pm 4,0)\), passing through \((8,2)\)

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex \(V(1,-2)\), focus \(F(1,0)\)

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{1}{2}\), vertices on the \(x\)-axis, passing through \((1,3)\)

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=\frac{4}{3}, \quad r \cos \theta=-3 $$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r \cos \theta=5 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices \(V(\pm 3,0), \quad\) asymptotes \(y=\pm 2 x\)

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric to the \(y\)-axis, and passing through the point \((2,-3)\)

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\)-intercepts \(\pm 2, \quad y\)-intercepts \(\pm \frac{1}{3}\)

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=3, \quad r=-4 \sec \theta $$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r \sin \theta=-2 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 10)\), asymptotes \(y=\pm \frac{1}{3} x\)

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric to the \(y\)-axis, and passing through the point \((6,3)\)

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\)-intercepts \(\pm \frac{1}{2}, \quad y\)-intercepts \(\pm 4\)

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=1, \quad r \sin \theta=-2 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(x\)-intercepts \(\pm 5, \quad\) asymptotes \(y=\pm 2 x\)

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r-6 \sin \theta=0 $$

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex \(V(-3,5)\), axis parallel to the \(x\)-axis, and passing through the point \((5,9)\)

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Horizontal major axis of length 8 , minor axis of length 5

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=4, \quad r=-3 \csc \theta $$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=2 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(y\)-intercepts \(\pm 2, \quad\) asymptotes \(y=\pm \frac{1}{4} x\)

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Vertex \(V(3,-2)\), axis parallel to the \(x\)-axis, and \(y\)-intercept 1

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertical major axis of length 7 , minor axis of length 6

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=\frac{2}{5}, \quad r=4 \csc \theta $$

Show that $$ x=a \cos t+h, \quad y=b \sin t+k ; \quad 0 \leq t \leq 2 \pi $$ are parametric equations of an ellipse with center \((h, k)\) and axes of lengths \(2 a\) and \(2 b\).

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ \theta=\pi / 4 $$

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertical transverse axis of length 10 , conjugate axis of length 14

Exer. 31-34: Find an equation for the set of points in an \(x y\)-plane that are equidistant from the point \(P\) and the line \(l\). $$ P(0,5) ; \quad l: y=-3 $$

Exer. 31-32: Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{l} x^{2}+4 y^{2}=20 \\ x+2 y=6 \end{array}\right. $$

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=\frac{3}{4}, \quad r \sin \theta=5 $$

Show that \(x=a \sec t+h, \quad y=b \tan t+k\) \(-\pi / 2

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Horizontal transverse axis of length 6, conjugate axis of length 2

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=4 \sec \theta $$

Exer. 31-34: Find an equation for the set of points in an \(x y\)-plane that are equidistant from the point \(P\) and the line \(l\). $$ P(7,0) ; \quad l: x=1 $$

Exer. 31-32: Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{l} x^{2}+4 y^{2}=36 \\ x^{2}+y^{2}=12 \end{array}\right. $$

Exer. 33-34: Find a polar equation of the parabola with focus at the pole and the given vertex. $$ V\left(4, \frac{\pi}{2}\right) $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$\frac{1}{3}(x+2)=y^{2}$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r^{2}\left(4 \sin ^{2} \theta-9 \cos ^{2} \theta\right)=36 $$

Exer. 31-34: Find an equation for the set of points in an \(x y\)-plane that are equidistant from the point \(P\) and the line \(l\). $$ P(-6,3) ; \quad l: x=-2 $$

Exer. 33-36: Find an equation for the set of points in an \(x y\)-plane such that the sum of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$ F(3,0), \quad F^{\prime}(-3,0) ; \quad k=10 $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$y^{2}=\frac{14}{3}-x^{2}$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r^{2}\left(\cos ^{2} \theta+4 \sin ^{2} \theta\right)=16 $$

Exer. 31-34: Find an equation for the set of points in an \(x y\)-plane that are equidistant from the point \(P\) and the line \(l\). $$ P(5,-2) ; \quad l: y=4 $$