Exam 7: Analytic Geometry

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A ________is the set of point (x, y) in a plane such that the difference in distances between (x, y) and two fixed points (called________ ) is a positive constant.

An equation of the form $y ^ { 2 } = 4 p x$ is given. Identify the vertex, value of p, focus, focal diameter, and endpoints of the latus rectum. - $4 y ^ { 2 } = 64 x$

Solve the system of equations. - (y-3=-8(x+3) -4x-6y=-22

Write the standard form of the equation of the hyperbola subject to the given conditions. -Vertices: $( - 5 , - 1 ) , ( - 5 , - 5 )$ ; Foci $( - 5 , - 3 + \sqrt { 15 } ) , ( - 5 , - 3 - \sqrt { 15 } )$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The standard form of an equation of a parabola with vertex (0, 0) and vertical axis of symmetry is________ .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A is the set of all points in a plane that are equidistant from a fixed line (called the________) and a fixed point (called the________ ).

Graph the parabola. Identify the directrix and focus. - $( x - 1 ) ^ { 2 } = - 4 ( y - 2 )$

Graph the ellipse. Identify the center and vertices. - $4 x ^ { 2 } + 25 y ^ { 2 } = 100$

Write the standard form of the equation of the hyperbola subject to the given conditions. -Corners of the reference rectangle: $( 2,10 ) , ( 2,0 ) , ( - 10,10 ) , ( - 10,0 )$ ; Horizontal transverse axis

Graph the ellipse. Identify the center and the endpoints of the minor axis. - $\frac { ( x + 1 ) ^ { 2 } } { 36 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$

The line segment perpendicular to the major axis, with endpoints on the ellipse, and passing through the center of the ellipse is called the axis.

Identify the foci and write equations for the asymptotes. - $9 ( y - 4 ) ^ { 2 } - 100 ( x + 4 ) ^ { 2 } = - 900$

Write the equation of the ellipse in standard form. Identify the center and foci. - $4 x ^ { 2 } + 9 y ^ { 2 } + 12 x - 108 y + 297 = 0$

Solve the problem. -The path of a comet can be modeled by the equation $\frac { x ^ { 2 } } { ( 1,169.7 ) ^ { 2 } } - \frac { y ^ { 2 } } { ( 26 ) ^ { 2 } } = 1$ where x and y are measured in AU (astronomical units). Determine the distance (in AU) at perihelion. Round to one decimal place.

Solve the problem. -The cross section of a cooling tower of a nuclear power plant is in the shape of a hyperbola, and can be modeled by the equation $\frac { x ^ { 2 } } { 576 } - \frac { ( y - 83 ) ^ { 2 } } { 2,304 } = 1$ where x and y are measured in meters. The base of the tower is located at y = 0, and the top of the Tower is 110 m above the base. Determine the diameter of the tower at the top. Round to the nearest Meter.

Graph the hyperbola. Identify the center and vertices. - $\frac { ( y - 1 ) ^ { 2 } } { 81 } - \frac { ( x + 1 ) ^ { 2 } } { 49 } = 1$

Solve the problem. -The quality of the images from an orbital telescope result from a large parabolic mirror, $3.8$ meters in diameter, that collects light from space. Suppose that a coordinate system is chosen so that the vertex of a cross section through the center of is located at $( 0,0 )$ . Furthermore, the focal length of the mirror is $56.8 \mathrm {~m}$ . Assume that $x$ and $y$ are in meters. Write an equation of the parabolic cross section of the mirror for $- 1.9 \leq x \leq 1.9$ .

Write the equation of the hyperbola in standard form. Identify the center and vertices. - $- 144 x ^ { 2 } + 9 y ^ { 2 } + 384 x + 36 y - 1,516 = 0$

Write the standard form of the equation of the hyperbola subject to the given conditions. -Vertices: (0, -6), (0, 6); Asymptotes: $y = - \frac { 3 } { 4 } x , y = \frac { 3 } { 4 } x$

Solve the problem. -The reflective property of an ellipse is used in lithotripsy. Lithotripsy is a technique for treating kidney stones without surgery. Instead, high-energy shock waves are emitted from one focus of an Elliptical shell and reflected painlessly to a patient's kidney stone located at the other focus. The Vibration from the shock waves shatters the stone into pieces small enough to pass through the Patient's urine. A vertical cross section of a lithotripter is in the shape of a semiellipse with the dimensions shown. Approximate the distance from the center along the major axis where the patient's kidney stone should be Located so the shock waves will target the stone. Round to two decimal places.