Exam 9: Analytic Geometry

Find an equation of the parabola described and state the two points that define the latus rectum. -Focus at (0, 4); directrix the line y = -4 A) $y ^ { 2 } = 4 x$ ; latus rectum: $( 9,2 )$ and $( - 9,2 )$ B) $x ^ { 2 } = 4 y$ ; latus rectum: $( 2,4 )$ and $( - 2,4 )$ C) $x ^ { 2 } = 16 y$ ; latus rectum: $( 8,4 )$ and $( - 8,4 )$ D) $x ^ { 2 } = 16 y$ ; latus rectum: $( 4,8 )$ and $( - 4,8 )$

Find a polar equation for the conic. A focus is at the pole. - $\mathrm { e } = 1$ ; directrix is parallel to the polar axis 1 above the pole

Solve the problem. -A spotlight has a parabolic cross section that is 6 ft wide at the opening and 2.5 ft deep at the vertex. How far from the vertex is the focus? Round answer to two decimal places.

Identify the equation without completing the square. - $2 x ^ { 2 } - 3 y ^ { 2 } + 3 x + 2 y + 2 = 0$

Graph the curve whose parametric equations are given. - $x = - \sec t , y = \tan t ; - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$

Find the asymptotes of the hyperbola. - $y ^ { 2 } - x ^ { 2 } = 9$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 2 t - 1 , y = t ^ { 2 } + 4 ; - 4 \leq t \leq 4$

Rotate the axes so that the new equation contains no xy-term. Graph the new equation. - $24 x y-7 y^{2}+36=0$

Solve the problem. - $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 4 } = 1$

Identify the conic that the polar equation represents. Also, give the position of the directrix. - $r = \frac { 5 } { 6 - 3 \sin \theta }$

Graph the equation. - $(x+2)^{2}=8(y-1)$

Solve the problem. -A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 8 feet across at its opening and is 2 feet deep at its center, at what position should the receiver be placed?

Find a polar equation for the conic. A focus is at the pole. - $e = 5$ ; directrix is perpendicular to the polar axis 3 to the right of the pole

Graph the curve whose parametric equations are given. - $x=2 \tan t, y=4 \sec t ; 0 \leq t \leq 2 \pi$

Find an equation for the hyperbola described. -center at $( 9,4 )$ ; focus at $( 7,4 )$ ; vertex at $( 8,4 )$

Find the vertex, focus, and directrix of the parabola with the given equation. - $(y-4)^{2}=-8(x+2)$

Graph the hyperbola. -Two recording devices are set 2,800 feet apart, with the device at point A to the west of the device at point B. At a point on a line between the devices, 400 feet from point B, a small amount of explosive is detonated. The Recording devices record the time the sound reaches each one. How far directly north of site B should a second Explosion be done so that the measured time difference recorded by the devices is the same as that for the first Detonation?

Name the conic. -

Find the vertex, focus, and directrix of the parabola. - $y^{2}=-12 x$ A) vertex: (0,0) focus: (-3,0) directrix: x=3 B) vertex: (0,0) focus: (3,0) directrix: x=-3 C) vertex: $( 0,0 )$ focus: $( 0 , - 3 )$ directrix: $y = 3$ 11ed81f4_181c_1451_a8e7_855e330b9a6b_TB7697_11 D) vertex: $( 0,0 )$ focus: $( 0 , - 3 )$ directrix: $y = 3$

Find the asymptotes of the hyperbola. - $\frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$