Math 5  Trigonometry  fall ’06  Chapter 6 Test Solutions.

 

1.      Find an equation for the parabola with vertex at (0,0) and directrix x = 2.  What is the focal diameter?  Construct a careful graph.
SOLN: with p = 2 we have the equation  and the focal diameter stretches from (-2, -4) to (-2, 4) and has length 8:
 

2.      Consider the conic described by . a.         has vertices at . b.         so the foci are at   c.       The eccentricity is   d.      Parametric equations:   e.       A careful graph showing the key features is shown at right:

 

3.      Consider the conic described by . a.        has vertices at   b.      Here  so the foci are at   c.       The asymptotes are along   d.      Parametric equations:   e.       A careful graph showing the key features is shown at right.

 

In 4  6, write an equation for the conic whose graph is shown.

4.      In the parabola below, the vertex is at (1,0) so the equation is .  Since the parabola passes through (9,2) we must have , so the parabola’s equation can be written

 

5.      The ellipse is centered at (1,2) with a = 4, b = 3:

6.   The hyperbola is centered at (1,0) , has vertices at a distance a = 2 from the center.  Thus we can write  .  Now plug in the coordinates of the given point:   Thus b = 1 and

7.      Write in standard form and sketch a graph: 
SOLN:  .  Since the left side of the equation is less than or equal to zero for all x and y, there is no graph in the real plane.

8.      Write in standard form and sketch a graph:
SOLN:  . 
This means the vertices of the hyperbola are at  and the asymptotes are along            

 

9.      Write an equation for the hyperbola with foci at  and asymptotes
SOLN:  The center is at  and c² = 2.  From the slopes of the asymptotes we know a = b so  which means that a = b = 1.  Thus the hyperbola is

10.  Consider the conic described by  

a.       Since B²  4AC = , equation describes an ellipse.

b.      The angle of rotation is  so we substitute
   &  whence  

c.       Sketch a graph for the equation.

d.      Where are the vertices in the xy coordinate system?
SOLN:  Rotate the vertices in the u-v system :  and , that is at  and .  On the minor axis the vertices are at  and .

11.  Consider the parametric curve given by                 a.       Construct a careful graph for the curve. b.      Eliminate the parameter θ to obtain an equation for this curve in rectangular coordinates.