91Ó°ÊÓ

Find the vertex, focus, and directrix of each parabola; find the center, vertices, and foci of each ellipse; and find the center, vertices, foci, and asymptotes of each hyperbola. Graph each conic. $$\frac{(x-3)^{2}}{16}-\frac{(y+4)^{2}}{9}=1$$

Find the first three terms of each recursively defined sequence. $$a_{1}=5, a_{n}=2 a_{n-1}$$

Find the vertex, focus, and directrix of each parabola; find the center, vertices, and foci of each ellipse; and find the center, vertices, foci, and asymptotes of each hyperbola. Graph each conic. $$9 x^{2}-y^{2}-36 x+6 y-9=0$$

Graph the path of the projectile that is launched at an angle of \(\theta\) with the horizon with an initial velocity of \(v_{0} .\) In each exercise, use the graph to determine the maximum height and the range of the projectile (to the nearest foot). Also state the time \(t\) at which the projectile reaches its maximum height and the time it hits the ground. Assume the ground is level and the only force acting on the projectile is gravity. \(\theta=42^{\circ}, v_{0}=315\) feet per second

Find the equation of the ellipse traced by a point \(P(x, y)\) that moves in such a way that the sum of its distances to (-3,1) and (5,1) is 10.

SATELLITE DISH A satellite dish has the shape of a paraboloid. The signals that it receives are reflected to a receiver that is located at the focus of the paraboloid. If the dish is 8 feet across at its opening and 1 foot deep at its vertex, determine the location (distance above the vertex of the dish) of its focus.

Find the term that contains \(b^{9}\) in the expansion of \(\left(a-b^{3}\right)^{8}\)

The fifth and seventh terms of an arithmetic sequence are -19 and \(-29 .\) Find the seventeenth term.

Use the Binomial Theorem to simplify the powers of the complex numbers. $$(1+2 i)^{5}$$

Show that \(\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}\right)=2^{n} .\) (Hint: Use the Binomial Theorem with \(x=1, y=1 .)\)