Analytic Geometry Answers

True or false,justify[no plane passes through the points(1,2,3),(1,-1,0)and(1,1,2)].

The line (x-1)/(3)=(y+1)/(4)=(z-2)/(3) passes through the point (4 2 3) check true or false

What surfaces in R3 are represented by the following equations? Z Z = 3 Y = 5 

D. Read and analyze the problem below. Solve and show complete solution.

Two stations, located at M(1.5, 0) and N(1.5, 0) (units are in km), simultaneously send sound

signals to a ship, with the signal traveling at the speed of 0.33 km/s. If the signal from N was

received by the ship four seconds before the signal it received from M, find the equation of the

curve containing the possible location of the ship.

A. Given the following information about the hyperbola, find both its standard and general form:

1. FOCI: (-4, 0) (4, 0); VERTICES: (-3, 0) (3, 0)

2. FOCI: (-3, -3) (-3, 13); VERTICES: (-3, 0) (-3, 10)

B. Given the standard form of the equation of hyperbola, find the following, then sketch the graph:

A. orientation

B. opening of branches

C. center

D. transverse axis

E. conjugate axis

F. distance of the foci

G. vertices

H. foci

I. co-vertices

J. asymptotes

1.(x+4)^2/9 − (y+3)^2/4 = 1

2.(y+2)^2/36 − (x−1)^2/49 = 1

C. Convert each general form to standard form of the equation of hyperbola: (10 points)

1. 9y^2 − 4x^2 − 18y + 24x − 63 = 0

2. 9x^2 - 4y^2 - 90x + 32y - 163 = 0

Read and analyze the problem below.

Solve and show your complete solution.

A semielliptical arch over a tunnel for a road through a mountain has a major axis of 100

feet, and its height at the center is 30 feet

(see figure) https://ibb.co/vLSmtVH

A) Find the standard form of the equation of the ellipse that represents the given problem.

B) Determine the height of the arch 5 feet from the edge of the tunnel.

A. Write BOTH the standard and general form of the equation of the specified ellipse. (20 points)

a. Vertices are at (5, -7) and (5, 5) and foci are at (5, -5) and (5, 3)

b. Vertices at (8, -1) and (0, -1) and whose minor axis has length 2.

(Check Examples 1 to 5 - basis for writing standard form)

(Check Examples 8 and 9 - basis for converting standard to general form)

B. Give the coordinates of the center, vertices, covertices, foci, orientation, lengths of major

and minor axis of each equation of ellipse. Then, sketch the graph. (20 points)

a. (x−3)^2/36 + (y+2)^2/100= 1

b. 9x^2 + 25y^2 − 18x + 100y − 116 = 0

Solve, show the complete solution and follow the format below for your final answers:

Orientation: Length of Major axis:

Center: Length of Minor axis:

Foci: Graph:

Vertices:

Co-vertices

Find the standard equation of the parabola which satisfies the given conditions.

1. Focus (−2, −5), directrix x = 6

2. Vertex (−4,2), focus (−4, −1)

3. Vertex (−8,3), directrix x = −10.5

4. Focus (7,11), directrix y = 4

Determine the focus and directrix of the parabola with the given equation. Sketch the

graph, indicate the focus, directrix, vertex, and axis of symmetry. (2 items, 20 points)

1. x^2 − 6x − 5y = −34

2. y^2 − 12x + 8y = −40

What is the vertex (h,k) given f= 2"\\sqrt{x-2}" +4?