Chapter 10

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{(x+1)^{2}}{16}+\frac{(y-4)^{2}}{8}=1$$

Find a polar equation that is equivalent to the given rectangular equation. $$x^{2}+y^{2}=25$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Ellipse; vertices \((2, \pi / 2)\) and \((8,3 \pi / 2)\)

In Exercises \(35-42,\) sketch the graph of the equation and label the vertex. $$y=4(x-1)^{2}+2$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$x^{2}+y^{2}-4 x-6 y+9=0$$

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$\frac{(x+5)^{2}}{4}+\frac{(y+2)^{2}}{12}=1$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{x^{2}}{25}+\frac{y^{2}}{18}=1$$

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$9 x^{2}+4 y^{2}+54 x-8 y+49=0$$

In Exercises \(35-42,\) sketch the graph of the equation and label the vertex. $$x=2(y-2)^{2}$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{(x-2)^{2}}{10}+\frac{(y+5)^{2}}{20}=1$$

Identify the conic section whose equation is given, and find its graph. If it is a circle, list its center and radius. If it is an ellipse, list its center, vertices, and foci. $$4 x^{2}+5 y^{2}-8 x+30 y+29=0$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{x^{2}}{9}-\frac{y^{2}}{15}=1$$

Identify the conic section and use technology to graph it. $$x^{2}+y^{2}+6 x-8 y+5=0$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(4 ;\) directrix; \(r=-2 \sec \theta\)

In Exercises \(35-42,\) sketch the graph of the equation and label the vertex. $$y=x^{2}-4 x-1$$

Find the equation of the hyperbola that satisfies the given conditions. Center (-2,3)\(;\) vertex (-2,1)\(;\) passing through \((-2+3 \sqrt{10}, 11)\)

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{y^{2}}{12}-\frac{x^{2}}{8}=1$$

Identify the conic section and use technology to graph it. $$x^{2}+y^{2}-4 x+2 y-7=0$$

Find a polar equation that is equivalent to the given rectangular equation. $$y=x-2$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(2 ;\) directrix: \(r=4 \csc \theta\)

In Exercises \(35-42,\) sketch the graph of the equation and label the vertex. $$y=x^{2}+8 x+6$$

Find the equation of the hyperbola that satisfies the given conditions. $$ \begin{aligned} &\text { Center }(-5,1) ; \text { vertex }(-3,1) ; \text { passing through }\\\ &(-1,1-4 \sqrt{3}) \end{aligned} $$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{(y-2)^{2}}{36}-\frac{(x-5)^{2}}{24}=1$$

Identify the conic section and use technology to graph it. $$4 x^{2}+y^{2}+24 x-4 y+36=0$$

Find a rectangular equation that is equivalent to the given polar equation. $$r=3$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(1 ;\) directrix: \(r=-3 \csc \theta\)

In Exercises \(35-42,\) sketch the graph of the equation and label the vertex. $$y=x^{2}+2 x$$

Find the equation of the hyperbola that satisfies the given conditions. Center (-5,1)\(;\) vertex (-3,1)\(;\) passing through \((-1,1-4 \sqrt{3})\)

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{(x+5)^{2}}{25}-\frac{(y-3)^{2}}{50}=1$$

Identify the conic section and use technology to graph it. $$9 x^{2}+y^{2}-36 x+10 y+52=0$$

Find a rectangular equation that is equivalent to the given polar equation. $$r=5$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(1 ;\) directrix: \(r=5 \sec \theta\)

Find the equation of the hyperbola that satisfies the given conditions. Center (-3,-5)\(;\) vertex (-3,0)\(;\) asymptote \(6 y=5 x-15\)

(a) What is the slope of the line through \((a, b)\) and \((c, d) ?\) (b) Use the slope from part (a) and the point \((a, b)\) to write the equation of the line. Do not simplify. (c) Show that the curve with parametric equations $$x=a+(c-a) t \quad \text { and } \quad y=b+(d-b) t$$ ( \(t\) any real number) is the line through \((a, b)\) and \((c, d) .\) [Hint: Solve both equations for \(t,\) and set the results equal to each other; compare with the equation in part (b).]

Identify the conic section and use technology to graph it. $$9 x^{2}+25 y^{2}-18 x+50 y=191$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(1 / 2 ;\) directrix: \(r=2 \sec \theta\)

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (0,0)\(;\) axis \(x=0 ;(2,12)\) on graph.

Identify the conic section and use technology to graph it. $$25 x^{2}+16 y^{2}+50 x+96 y=231$$

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Eccentricity \(4 / 5 ;\) directrix: \(r=3 \csc \theta\)

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (0,1)\(;\) axis \(x=0 ;(2,-7)\) on graph.

Use Exercise 44 to find a parameterization of the line segment joining the two points. Confirm your answer by graphing. $$(-6,12) \text { and }(12,-10)$$

Find the equation of the ellipse that satisfies the given conditions. Center (2,3)\(;\) endpoints of major and minor axes: (2,-1), (0,3),(2,7),(4,3).

Find a rectangular equation that is equivalent to the given polar equation. \(r=\sec \theta[\text {Hint}:\) Express the right side in terms of cosine.]

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Hyperbola; vertical directrix to the left of the pole; eccentricity \(2 ;(1,2 \pi / 3)\) is on the graph.

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (1,0)\(;\) axis \(x=1 ;(2,13)\) on graph.

Use Exercise 44 to find a parameterization of the line segment joining the two points. Confirm your answer by graphing. $$(14,-5) \text { and }(5,-14)$$

Find the equation of the ellipse that satisfies the given conditions. Center (-5,2)\(;\) endpoints of major and minor axes: (0,2), (-5,17),(-10,2),(-5,-13).

Find the polar equation of the conic section that has focus (0,0) and satisfies the given conditions. Hyperbola; horizontal directrix above the pole; eccentricity \(2 ;(1,2 \pi / 3)\) is on the graph.

In Exercises \(43-54\), find the equation of the parabola satisfying the given conditions. Vertex (-3,0)\(;\) axis \(y=0 ;(-1,1)\) on graph.

Find the equation of the ellipse that satisfies the given conditions. Center (7,-4)\(;\) foci on the line \(x=7 ;\) major axis of length \(12 ;\) minor axis of length 5.