91Ó°ÊÓ

Sketch the graphs of the given curves and compare them. Do they differ and if so, how? (a) \(x=t, \quad y=t^{2}\) (b) \(x=\sqrt{t}, \quad y=t\) (c) \(x=e^{t}, \quad y=e^{2 t}\)

Calculus can be used to show that the area of the ellipse with equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(\pi\)ab. Use this fact to find the area of each ellipse. $$5 x^{2}+y^{2}=5$$

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. $$\frac{(x-1)^{2}}{4}+\frac{(y-5)^{2}}{9}=1$$

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$x^{2}-4 y=0$$

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. circle with center (7,-4) and radius 6

Convert the rectangular coordinates to polar coordinates. $$(\sqrt{5}, \sqrt{10})$$

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

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. $$\frac{(x-2)^{2}}{16}+\frac{(y+3)^{2}}{12}=1$$

In Exercises \(29-34,\) find the latus rectum of the parabola whose equation is given. [Hint: Examples 3 and 4 may be help. ful in Exercises \(29-30.1\) $$y^{2}+12 x=0$$

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. circle with center ( 9,12 ) and radius 5